Quantum Interferometry with Multiports: Entangled Photons in Optical Fibers

Dissertation

zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften eingereicht von

Mag. phil. Mag. rer. nat. Michael Hunter Alexander Reck

im Juli 1996

durchgeführt am Institut für Experimentalphysik, Naturwissenschaftliche Fakultät der Leopold-Franzens Universität Innsbruck unter der Leitung von o. Univ. Prof. Dr. Anton Zeilinger

Diese Arbeit wurde vom FWF im Rahmen des Schwerpunkts Quantenoptik (S06502) unterst¨utzt. Contents

Abstract 5

1 Entanglement and Bell’s inequalities 7 1.1 Introduction ...... 7 1.2 Derivation of Bell’s inequality for dichotomic variables ...... 10 1.3 Tests of Bell’s inequality ...... 14 1.3.1 First experiments ...... 14 1.3.2 Entanglement and the parametric downconversion source .16 1.3.3 New experiments with entangled photons ...... 17 1.4 Bell inequalities for three-valued observables ...... 18

2 Theory of linear multiports 23 2.1 The beam splitter ...... 24 2.2 Multiports ...... 25 2.2.1 From experiment to matrix ...... 26 2.2.2 From matrix to experiment ...... 27 2.3 Symmetric multiports ...... 32 2.3.1 Single-photon eigenstates of a symmetric multiport .... 33 2.3.2 Two-photon eigenstates of a symmetric multiport ..... 33 2.4 Multiports and quantum computation ...... 35

3 Optical fibers 39 3.1 Single mode fibers ...... 39 3.1.1 Optical parameters of fused silica ...... 40 3.1.2 Material dispersion and interferometry ...... 43 3.2 Components of fiber-optical systems ...... 46 3.3 Coupled waveguides as multiports ...... 47

4 Experimental characterization of ﬁber multiports 49 4.1 A three-path Mach-Zehnder interferometer using all-ﬁber tritters .49 4.1.1 Experimental setup ...... 49 4.1.2 Theoretical description ...... 52 4.1.3 Experimental results ...... 55

1 2

4.2 Two-photon interferences in optical ﬁber multiports ...... 56 4.2.1 Introduction ...... 56 4.2.2 Theoretical description ...... 56 4.2.3 Experimental results ...... 58

5 Two-photon three-path interference 63 5.1 Energy entanglement in three-path interferometers ...... 63 5.2 Multipath interferences ...... 64 5.3 Experiment ...... 70 5.3.1 The experimental setup ...... 70 5.3.2 Polarization and path length adjustment ...... 79 5.3.3 Detection system ...... 84 5.3.4 Calibration of phase settings ...... 88 5.3.5 Data acquisition ...... 90 5.3.6 Temperature drifts ...... 91 5.4 Experimental results ...... 93 5.4.1 Description of data ...... 93 5.4.2 Data analysis ...... 100 5.5 Interpretation ...... 105 5.5.1 Bell’s inequalities ...... 105 5.5.2 Classical picture vs. quantum picture ...... 106

6 Conclusions and Outlook 111

A Devices 115 A.1 Parametric downconversion crystals ...... 115 A.1.1 KDP and LiIO3 ...... 115 A.1.2 Beta Barium Borate (BBO) ...... 116 A.2 Optical ﬁbers and tritters ...... 117 A.3 Lasers ...... 119 A.3.1 Argon-Ion laser ...... 119 A.3.2 HeNe laser ...... 120 A.3.3 Diode laser ...... 121 A.4 Single-photon detectors ...... 121 A.5 Control and detection electronics ...... 122 A.6 Miscellanea ...... 123 A.6.1 Electro-mechanic shutter ...... 123 A.6.2 Parallel printer port as programmable TTL output .... 123 A.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 ...... 133 C.2 Mathematica package ...... 136

D Wave packets in multiports 141 D.1 A single photon in a multiport ...... 141 D.2 Photon pairs in multiports ...... 143 D.3 Material dispersion and interference ...... 144

EPhotographing Type-II downconversion 149 4 Abstract

The present thesis is the result of theoretical and experimental work on the physics of optical multiports. The theoretical results show that multiport interferometers 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 (tritters) 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 inequalities. 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 entanglement 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 experiment 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 mechanical 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.

Schr¨odinger’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. ...Iftwoseparatedbodies,eachbyitselfknownmaximally,entera 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 [Schrödinger35]

Schr¨odinger 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 individual 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) Schrödinger did not use the word entanglement. He used the German word ‘Verschränkung’, which in English would be better rendered as entwinement. The original Version is: “Maximale Kenntnis von einem Gesamtsystem schließt nicht notwendig maximale Kenntnis aller seiner Teile ein, auch dann nicht, wenn dieselben völlig voneinander abgetrennt sind und einander zur Zeit gar nicht bee- influssen. ...WennzweigetrennteKörper, die einzeln maximal bekannt sind, in eine Situation kommen, in der sie aufeinander einwirken, und sich wieder tren- nen, dann kommt regelmäßig das zustande, was ich eben Verschränkung unseres Wissens um die beiden Körper nannte.” 9

Bohr, the founder of the “Copenhagen interpretation” of quantum mechanics, 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 considered a matter of interpretation. Physicists were more concerned with the application 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 experiment 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 dichotomic variables

Here we will present a simple derivation of the Bell inequality based on arguments 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.

+ + ϕ S ϕ' - -

Figure 1.1: Schematic of a Bell inequality experiment. The source emits particles that have two states. Analyzers on each side are described by black 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 consider 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,andγγ 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 − − + − + + γ − + α Figure 1.2: Cube representing hidden variables assigned to classes of particles. The hidden variables determine the results for measurements at analyzer settings α, β,andγ on one side, and, because of perfect correlations, for the corresponding settings α, β,andγ on the other side. Quantum mechanics predicts a violation of Bell’s inequality for particles that 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 √1 |Ψ = (|H1|H2 −|V 1|V 2) . (1.8) 2 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 polarization 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 settings α on one side and β on the other are sinusoidal functions of the diﬀerences of the angular settings of the analyzers: 1 N ++(α, β)= cos2(α − β), (1.9) QM 2 1 N +−(α, β)= sin2(α − β). QM 2

If we choose the angles {α = α =0◦, β = β =30◦,andγ = γ =60◦} the QM predictions are 3 N ++(α, β)= , (1.10) QM 8 2 N ++(α, γ)+N +−(β,γ)= . (1.11) QM QM 8 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, experimentally 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 inequality 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 function of 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 analyzers. 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 inequality 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 assumption 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 energyhω ¯ 1 andhω ¯ 2. 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 treatment 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 polarization 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 momentum 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 overlapof the modes of two PDC crystals led to in- 17 terferences in the singles-rates that could be modulated by the phases and the degree of fundamental distinguishability [Zou91]. Experiments in Innsbruck demonstrated 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 interferometry 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 implemented 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 proposed 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 swapping) [Zukowski93]. The transfer of a quantum state by use of the EPR correlations 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 observables

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 ¯h spin > 2 particles. Bell inequalities for such higher-dimensional spin systems have been analyzed by Garg, Mermin, and Peres. Quantum mechanics is in contradiction 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 quantum physics. The Kochen-Specker theorem gives a mathematical proof of the impossibility of assignment of elements of physical reality to unmeasured properties 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 2 2 2 2 particle J = Jx + Jy + Jz , Jx, the angular momentum in x-direction, and Jy,the 2 angular momentum in y-direction are observables. J and Jx commute, so we can 2 2 measure J together with Jx. We also notice that J and Jy commute. So we can 2 measure J together with Jy. The assumption of noncontextuality implies that 2 the results of the measurement of J , be it alone, together with Jx, or together 19

with Jy, should be the same. Kochen and Specker showed that noncontextuality 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.

+ + (φ φ ) (φ φ ) o 1, 2 S ''1, 2 o - - Φ Φ' Figure 1.3: Schematic of a source for a three-state Bell inequality experiment. Analyzers on each side are described by black boxes with a knob that can 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 separated 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 settings 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 + + −

A ο ο + − − − − + − ο + + ο − B C Figure 1.4: Hidden variable assignment deﬁnes subsets of the ensemble of particle pairs with three-valued measurement outcomes. Only 27 domains of the hidden variable space are nonempty. The hidden variables determine the results for measurements at analyzer settings A, B,and C on one side, and, because of perfect correlations, for the corresponding settings A, B,andC on the other side. The hidden variables determine the outcome of a detection event {+, −, ◦} as function of the setting. The quantum mechanical prediction for particles that are members of the shaded 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 = {(++), (+◦), (+−)})isgivenby 1 N QM ∝ [3 + 2 cos (φ + φ + χ) (1.17) xy 9 1 1 − +2cos(φ2 + φ2 χ) − − +2cos(φ2 + φ2 φ1 φ1 + χ)] 22

{ 2π − 2π } with the ﬁxed parameter χ = 3 , 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 ◦− −◦ − − − ++, , φ1 = φ1 α φ2 = φ2 + α ◦◦ − − − − − ,+ , + φ1 = φ1 + α φ2 = φ2 α −− ◦ ◦ − − , +, + φ1 = φ1 φ2 = φ2

≡ 2π with α 3 . 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 ≡ π analyzers (γ 5 ):

(φ1,φ2) (φ1,φ2) 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 −0.2322411 which 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 quantum 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:

2 2 ||Ψ1| = |U |Ψ0| . (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 (k1,k2). We will use the following representation of the beam splitter matrix:       iφ  k   sin ωecos ω   k1   1  =     . (2.3) − iφ k2 cos ω e sin ω k2

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.

k1 k'2

k2 ϕ k'1

Figure 2.1: The beam splitter coherently transforms two input modes (k1,k2) into two output modes (k1,k2).Theparameterω describes the reﬂectivity and transmittance of the device. The phase ϕ shifts the relative phase 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

k k' 1 ω 1

Τ=1/2 Τ=1/2

ϕ -ω k2 k'2 Figure 2.2: Mach-Zehnder realization of a beam splitter with variable reflectivity. The internal phase ω changes the reflectivity of the 4-port device. 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 multiports 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 1' 2 2' 3 U(N) 3'

N N'

Figure 2.3: A multiport device consisting of beam splitters, mirrors, and phase shifters can be viewed as a black box. It will be described by a unitary matrix 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 matrices 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)   See Sei   ain(ext)    =     . (2.5) aout(int) Sie Sii ain(int)

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: −1 aout = See + Sei (Γ − Sii) Sie ain. (2.7)

Thus we have can easily calculate the unitary matrix of the system out of our knowledge of the topology.

−1 U = See + Sei (Γ − Sii) Sie. (2.8)

This method gives a simple formula for the unitary matrix of a complex interferometer 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 successively 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 splitters for atom de Broglie 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 general 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 application 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 matrix except that the four matrix elements with the indices pp, pq, qp,andqq have been replaced by the corresponding beam splitter matrix elements. This matrix 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)forq = 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 − 1) 0  · · ···   U(N) TN,N−1 TN,N−2 TN,1 =   . (2.9)     0 eiα 29

T32

T31 T32

T21 T31 T32

α1 α 2 T21 α 3 T31 T32

Figure 2.4: Example for the implementation of the algorithm to construct a 3 × 3 unitary matrix. On the left hand two-dimensional unitary transforms are performed on two-dimensional subspaces, successively making the complex matrix elements (∗) zero. On the right hand side the experiment is successively built up from the corresponding beam splitters, represented by four-port devices. In the last step phase shifters are inserted to make the diagonal matrix elements equal to one. The experimental setup equivalent to the original unitary matrix is the device in the last line 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)

-α1 † α T - 2 † † 21 α T32 T31 - 3

† Figure 2.5: Three beam splitter devices Tpq (operated in reverse) and three additional phase shifters αi are enough to build any 3 × 3 unitary matrix. Notice that because of operation in reverse the individual devices’ phase shifts are now at the input ports and the ﬁnal phase shifters αi at the output 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:

−1 U(N)=(TN,N−1 · TN,N−2 ···T2,1 · D) . (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 interferometers 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. A suitable beam splitter is at each crossing point of the beams. Phase shifters are at one input of each beam splitter and at the outputs (1 ...N)ofthe multiport. Each diagonal row of beam splitters performs a transformation reducing 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 matrix H. We denote the eigenvectors as |v1 ...|vN and the corresponding eigenvalues with {h1,h2,...hN },asetofN 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 for the second, and vi|ψ for the i-th. If the i-th detector fires, we assign the measurement 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 corresponding 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 elements 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 1 probability of detecting a photon at any output is N . 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 2π N-th root of unity γN =expi N [Zukowski94], we can write the matrices for a symmetric 2N multiport as follows:

1 − − U (N) = √ γ(m 1)(k 1). (2.13) mk N N The beam splitter matrix in this form is   1  11 B = √   . (2.14) 2 1 −1

In analogy with the beam splitter the 3×3 symmetric multiport is called a tritter. The canonical form of the tritter matrix is    11 1   1   √  2  ≡ i 2π/3 T =  1 αα with α e (2.15) 3   1 α2 α 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 can calculate 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 × N Hilbert 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 matrices:    11 1 1       1  1 −11−1  ⊗   B B =   (2.21) 2  − −   11 1 1    1 −1 −11

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 +1 111−1 1001 −1 1 −1 −1 −1 01−10

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: 1 |Ψ± = √ (|a|b±|b|a) , (2.22) 2 1 |Φ± = √ (|a|a±|b|b) . (2.23) 2 These maximally entangled basis states are called Bell states and play an important 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 1 111 111 01 11−1 −11−1 −1 −1 2 −1 −1 −1 −1 −1 −1 −1 −1 01 1−1 −11 −11−1 01 1−11−1 −1 −11 √ √ +i 01+3 −1 − 301−101−1 √ √ 00 01+311−1 − 3 −1 −1 √ √ −i 00 01− 311−1+ 3 −1 −1 √ √ 01− 3 −1+ 301−101−1

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:   iφ  sin ωecos ω    . (2.25) cos ω −eiφ sin ω 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.The operation of the controlled-not gate can be written as a simple matrix in the {|a|a, |a|b, |b|a, |b|b} basis:    1000        0100   C =   . (2.26)    0001   0010

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 asatensorproductofthemultiportmatrices:

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 contradiction. Therefore a controlled-not gate can not be built from linear multiports.

Linear unitary multiports can be used for state preparation of single quanta in quantum communication and quantum computation. They provide a handy formalism for the description of complicated interferometric experiments and can be easily generalized to describe multifrequency radiation. In the next chapter we will see how multiports can be realized using optical ﬁbers. 38 Chapter 3

Optical ﬁbers

Optical fibers are in wide spread use as the main carriers of high-speed earth- based communication networks. Extensive studies have been made of the use of fibers in communications technology but little research has been published on the use of optical fibers in experiments on the foundations of quantum mechanics.

A notable exception is the group at DRA, Malvern. Rarity and Tapster have measured nonclassical interferences in fiber interferometers [Rarity92]. The aim of their research has been to demonstrate a quantum cryptography scheme based on Bell’s inequalities. By using the optical fibers they were able to achieve aviolationofaBellinequalityovera4kmlongopticalfiberonaspoolinthe 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 2 − 2 NA = ncore ncl . (3.1) The same expression applies to any axis-symmetric graded-index fiber if we assume 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 description 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) r V ≡ 2π NA (3.2) λ0 that gives the radius of the core r in units of wavelength λ0.IfV<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: A λ2 n(λ)=1+ j . (3.3) 2 − 2 j λ λj

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 material and the vacuum wave vector w k(ω)=n(ω) k = n(ω) . (3.4) vac c

When we have narrow bandwidth light with center frequency ω0 we can use the Taylor expansion of n around ω : 0 ∂n 1 ∂2n n(ω)=n(ω )+ (ω − ω )+ (ω − ω )2. (3.5) 0 ∂ω 0 2 ∂ω2 0 ω0 ω0

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 groupvelocity and groupindex are c vg(λ)= , (3.6) ng(λ) ∂n n (λ)=n(λ) − λ . (3.7) g ∂λ 42

1.480 ng 1.475

1.470

1.465

1.460 n 1.455 Index of refraction 1.450

1.445 0.6 0.8 1.0 1.2 1.4 1.6 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 ∂2 k ∂ 1 Dω = 2 = , (3.9) ∂ω ∂ω vg 2πc D = − D . (3.10) λ λ2 ω In terms of wavelengths they are λ3 ∂2 n Dω =2π 2 2 , (3.11) c ∂λ 2 −λ ∂ n ∂ 1 Dλ = 2 = , (3.12) c ∂λ ∂λ vg 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.

-50

-100

-150

[ps/(km nm)] -200 λ D -250

-300

0.6 0.8 1.0 1.2 1.4 1.6 Wavelength [µm]

Figure 3.2: Wavelength dependence of the group velocity dispersion coef- ﬁcient calculated from the Sellmeier formula for fused silica. Between the regions of normal dispersion Dλ < 0 and anomalous dispersion Dλ > 0 the dispersion coeﬃcient is zero. This is the telecommunication wavelength λ =1.3 µm. High data rates are achievable because pulse broadening is minimal at this wavelength. 44

The parameter describing the dispersion in the Mach-Zehnder interferometer 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 λ2 = − D . (3.13) ∂ω2 2πc λ

ps The units for the dispersion coeﬃcient Dλ listed in data sheets are nm km , 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

ps Dλ [ nm km ] -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 groupvelocity dispersion ( λ =1.3 µm) pulse broadening is minimal (Fig. 3.2). This is the wavelength most often used in telecommunication 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 (ω − ω )2 f(ω) ∝ exp − 0 (3.14) σ2 the power or probability distribution is the square of this function: 2(ω − ω )2 P (ω) ∝ exp − 0 . (3.15) σ2

In the narrow-bandwidth limit (σ ω0) the power or probability distribution as function of wavelength λ is: 2(λ − λ )2 P (λ) ∝ exp − 0 , (3.16) ∆λ2

λ2 with λ ≡ 2πc and ∆λ ≡ 0 σ. Then the full-width-half-maximum filter pa- 0 ω0 2πc 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 filter bandwidths (full-width-half-maximum) given in catalogues. They are related to bandwidth σ (e−1 width of the Gaussian frequency amplitude distribution) by: 2 πc σ = 2 ∆λFWHM. (3.17) ln 2 λ0

1,0 1 nm 0,9 2 nm 0,8 0,7 3 nm 0,6

Visibility 0,5 λ = 789 nm 0,4 0,3

1,0 0,9 1 nm 0,8 0,7 0,6 2 nm Visibility 0,5 λ = 633 nm 0,4 3 nm 0,3 0,0 0,5 1,0 1,5 2,0 ∆x [m]

Figure 3.3: Calculated single photon interference visibility in a fiber interferometer at λ = 633 nm and 789 nm with three different bandwidths ∆λFWHM as function of the path-length difference. The lower group velocity dispersion at longer wavelengths gives a considerably higher visibility.

The visibility of interference fringes calculated using the material parameters 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 groupvelocity 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 between polarization modes by stress or dopant-induced birefringence. The fiber used in our experiment had circular (symmetric) cores. Manual polarization controllers 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 evanescent 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 singlemode 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   cos(KL) −i sin(KL)   k1   1  =     . (3.18) − k2 i sin(KL)cos(KL) k2 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 × 3coupler which is shown in Fig. 3.5.

In the following experiments we analyzed the properties of 3 × 3fibercou- 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 second 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 ﬁber cores are brought close together in the fused coupling region. The distance between the cores is of the order of the core diameter. The evanescent wave from one core can excite the other cores. After a distance and wavelength dependent coupling length the power is symmetrically split between the cores.

1,0 T R 0,8

0,6

0,4

0,2 Normalized transmission

0,0 600 650 700 750 800 Wavelength [nm]

Figure 3.5: Measured wavelength dependence of matrix elements of the 3 × 3 integrated fiber coupler. The intensity transmission into the same output T and into the other outputs R wasmeasuredasfunctionofthe wavelength. This integrated fiber coupler was used in the 789 nm fiber interferometer. At this wavelength the intensity splitting ratio is 1/3 into each 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 3×3fibercouplers. 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 × 3couplers[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

RC-generators Amplifiers Oscilloscope Trig < < Trig ϕ1

I0 I1 ϕ2 HeNe-Laser I2 Tritter 1 Tritter 2 I3

Piezo Polarization FC/PC- Photodiode phase shifter controller connector Figure 4.1: Schematic of the three-path interferometer experiment. The tritters are three-way integrated ﬁber couplers. Polarization controllers were used to manually adjust the polarization in all three arms for maximal visibility. Piezo tubes were used to modulate the interferometer phases φ1 and φ2 in two of the three interferometer arms. The photodiode signals 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 24 mm. 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

1.0 radius = 24mm 0.8 loops = 1, 2, 3

0.6

0.4

Retardation [rad] 0.2

0.0 0.6 0.8 1.0 1.2 λ [µm]

Figure 4.2: Calculated induced retardation in the manual polarization controller. A singlemode optical ﬁber with cladding diameter 125 µm is wrapped around a disk of radius r =24mm. This induces an optical retardation as a function of the number of loops on the disk (1, 2, 3) and the wavelength (λ). Any polarization can be set by using three tilted disks if the retardation on each disk lies between π/3 and 2π/3.Forλ = 633 nm one 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 40 mm, length 18 mm, and wall thickness 1 mm contract when a high voltage is applied. The specifications were about 13 µm 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 0.08 µm/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 ≤ 1s. 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 input 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 out out out in (E1 ,E2 ,E3 )=MK . The three-path interferometer can be described by the multiport matrix    eiφ1 00      iφ2  M = T  0 e 0  T. (4.1)   00eiφ3

T is the tritter matrix of eq. (2.15)andφ1 ...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 | out|2 In = En (4.2) I = 0 [3 + 2 cos(φ − φ + χ )+2cos(φ − φ + χ )+2cos(φ − φ + χ )] . 9 1 2 n 2 3 n 3 1 n

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 I(φ ,φ ,φ )= 0 [3 + 2 cos(φ − φ )+2cos(φ − φ )+2cos(φ − φ )] . (4.3) 1 2 3 9 1 2 2 3 3 1 Sometimes it is useful to rewrite the function in eq. (4.3) as a product of cosine functions: I φ − φ φ − φ φ − φ I(φ ,φ ,φ )= 0 1+8cos 1 2 cos 2 3 cos 3 1 . (4.4) 1 2 3 9 2 2 2 53

1 2π 0.5

0 0 π

π φ −φ23 φ −φ 13 2π 0 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 pattern are 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 maximum to minimum as this phase is changed. The curve shows maximal visibility (see Fig. 4.4).

0.8

0.6

0.4

0.2

0 0 1 2 3 4 5 6

φ12 = −φ [rad ] Figure 4.4: The calculated three-path Mach-Zehnder interferences show maximum visibility when φ2 = −φ1 is varied. The fringes observed are non-sinusoidal. The parts of the curve where the slope is steeper than that of two path interferences show higher phase sensitivity as discussed in the text. 54

The phase setting φ2 = −φ1 also gives the maximal sensitivity S,whichis deﬁned as the derivative of the output intensity with respect to a phase [Sheem81], 1 dI S = . (4.5) I0 dϕ

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

4.1.3 Experimental results

ϕ [rad] ϕ [rad] ϕ [rad] 2 2 2 CH1 02π/3 4π/3 CH2 02π/3 4π/3 CH3 02π/3 4π/3

44 2π

[V] π 1 4 /3 ϕ ϕ 1 36 [rad] 2π/3 32 piezo voltage 0 28

28 32 36 40 44 28 32 36 40 44 28 32 36 40 44 piezo voltage ϕ [V] 2

Figure 4.5: Experimental data of the three-path Mach-Zehnder interferometer shown as gray-scale plots (minimum white, maximum black). The threeoutputintensities(CH1...CH3)areshownasfunctionsofthetwo relative phase settings. The visibility of the interferences was approximately 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 frequencies (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 a tritter. 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).

kl Feynman’s rules can be used to obtain the probability Pij 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

(cl)kl | |2 | |2 P ij = MikMjl + MilMjk . (4.6)

If, however, the particles are indistinguishable, as is the case with two photons 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- (qm)kl come. For indistinguishable particles the quantum probability P ij may show destructive interference:

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 indistinguishable particles) and P∞ for τin →∞(classically distinguishable). ∗ 2 2 − ω t kl P∞ P0 2Re[(MikMjl) (MilMjk)] − d c V e 8 . ij = = 2 2 (4.8) P∞ |MikMjl| + |MilMjk|

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 = 200 µm would already halve the visibility.

4.2.3 Experimental results

In our experiment we used correlated photons of equal wavelength from a parametric downconversion crystal. The degenerate mode was selected by irises (see 1The so deﬁned visibility describes the two-photon interference and is diﬀerent from the one used for interference fringes. 59

Interference filter 702nm Polarization Connector RG-630 filter controller Si-APD

∆τ x=c in

KD*P- counters

crystal coincidence Tritter Ar+-laser (351.1 nm)

Figure 4.7: The schematic of the two-photon interference experiment in a fiber tritter. The entangled modes from the crystal were coupled into two inputs of the fiber tritter. A coincidence unit recorded the coincidences between all three pairs of outputs as the time-lag between the arrival times of 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 operation 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    0.80 0.43 0.43      (3)  i0.83π i1.41π  Mexp =  0.43 0.80e 0.43e  , (4.9)   0.43 0.43ei1.41π 0.80ei0.83π with errors in the moduli and phases less than 2%. The maximum possi- 12 ± ble visibilities calculated from Eq. (4.8) are then V12 =(46.0 3.0)% and 12 12 ± V13 = V23 =(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 objectives on translation stages used in the two-photon interference experiment in a fiber 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.

12 12 12 V12 V13 V23 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

2.4 2.2 2.0 1.8 1.6 1&2

] in 10s 1.4 3 1.2 1.1 1.0 2&3 Counts [10 0.9

0.8

0.7 1&3

-400 -200 0 200 400 600 Position ∆ x [µm]

Figure 4.9: Photon antibunching at the outputs of a fiber tritter can be modelled as a Gaussian dip. The coincidence counts in 10 s between the output ports of the integrated fiber tritter are shown as a function of the position of the input coupler (see Fig. 4.7). The count rates were corrected for a linear drift of the effective laser power, with the actual coincidence rates at ∆x = −400 as reference. The rates show dips as the path length difference goes to zero. The visibilities of the dips were 12 ± 12 ± 12 ± V12 =(40 1)%, V13 =(22 2)%,andV23 =(20 2)%, where the upper indices denote the inputs and the lower indices the measured outputs of the tritter. The width of the coincidence dip between outputs 1 and 2 was ∆xFWHM = 300 ± 10 µm, corresponding to a coherence length of lc = 255 ± 9 µm. The width of 300 µm corresponds to a time-delay of 1ps in the vacuum. 62 Chapter 5

Two-photon three-path interference

5.1 Energy entanglement in three-path interferometers

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,andl 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 function 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 dropthe 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: 1 Ψ = (Ψ +Ψ +Ψ +Ψ +Ψ +Ψ +Ψ +Ψ +Ψ ) . (5.1) tot 9 ss sm sl ms mm ml ls lm ll

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: Ψ ∗ P12(t1,t2)=ΨtotΨtot. (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 s λ i=789nm m λ p=351nm l' Coinc.

λ =633nm s s' m' Figure 5.1: Two modes from the parametric downconversion crystal are coupled into two 3-path interferometers. Two single-photon detectors register coincidence counts between two separated outputs. When the path length differences l-l’, m-m’, and s-s’ are exactly equal, second order interferences 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 consider the energy correlations: ∞∞ | − † † | Ψ = ∆(ω + ω ωp)â1(ω)â2(ω ) 0 . (5.4) 0 0 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 apertures. We assume Gaussian filters g and g with center frequencies ω0 and ω0,and bandwidths σ and σ. The two-photon wave function thus changes to: ∞∞ | − † † | Ψ = g(ω)g (ω )∆(ω + ω ωp)â1(ω)â2(ω ) 0 dω dω . (5.5) 0 0 1Since the polarizations of the photons emitted in type-I downconversion process are the same, we assume only one polarization mode in the following discussion. 66

In the narrow bandwidth limit (σ ω0, σ ω0) the integral can be extended to inﬁnity and we ﬁnd that the time-dependent wave function describes a nonfactorizable state:    2 2 σ x x Ψ(x, x,t,t) ∝ ei(kx−ωt) ei(k x −ω t ) exp − ( − t) − ( − t) . (5.6)  4 c c 

For the moment we neglect dispersion and assume equal optical path-length differences 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 interferometers 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:

Ψ ∝ 2 2 2 2 P12(a, b, τ) 3[F (0)] +[F (a)] +[F (b)] +[F (a + b)] (5.10) +4cos(ka) {F (0) F (a)+F (a + b) F (b)} +4cos(kb) {F (0) F (b)+F (a) F (a + b)} +4cos(k (a + b)) {F (0) F (a + b)+F (a) F (b)} +2cos(k (2 a + b)) F (0) F (b) +2cos(k (a +2b)) F (0) F (a) +2cos(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: σ2 σ2 F (u) ≡ exp − [u + cτ]2 +exp − [u − cτ]2 (5.11) 4 4 67 which for u = 0 reduces to σ2 (cτ)2 F (0) ≡ 2exp − . (5.12) 4

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 Ψ∗Ψ dτ ∝ (5.13) − T/2            3/2 +cos(2 ak) +cos(2 bk) +cos(2( a+ b) k) H(0)   ss−ss ss−mm mm−ll ss−ll mm−mm mm−ss ll−mm ll−ss ll−ll

+H(2 a)+H(0)G(2 a) sm−sm,sm−ms ms−sm,ms−ms

+H(2 b)+H(0)G(2 b) ml−ml,ml−lm lm−ml,lm−lm

+H(2( a + b)) + H(0) G(2 (a + b) k) sl−sl,sl−ls ls−sl,ls−ls

+2cos(( a +2b) k)H(a)G(a) ml−ss,lm−ss ss−ml,ss−lm

+ 2 cos((2 a + b) k)H(b)G(b) ll−ms,ll−sm ms−ll,sm−ll − +2cos(( a b) k)H( a + b)G(a + b) mm−sl,sl−mm ls−mm,sl−mm        +2cos(ak)  2H(a)G(a) +H( a +2b)G(a)+H( a)G(a +2b) sm−ss,ms−ss ml−sl,ls−ms mm−sm,mm−ms,... lm−sl,lm−ls,... 68         +2cos(bk)  2H(b)G(b) +H(2 a + b)G(b)+H( b)G(2a + b) lm−mm,ml−mm sl−sm,ms−sl ll−ml,ll−lm,... ls−sm,ls−ms,...    +2cos((a + b) k) 2H(a + b)G(a + b) sl−ss,ls−ss ll−sl,ll−ls,...    − −  +H( a b)G(a + b)+H( a + b)G(a b) . ml−sm,ml−ms lm−sm,lm−ms,...

The expression is a sum of cosines multiplied with Gaussians and error functions: (σx/c)2 G(x) ≡ exp − , (5.14) 8 √ 2π σ (x/c − T ) σ (x/c + T ) H(x) ≡ Erf , , (5.15) cσ 23/2 23/2 y 2 2 Erf(x, y) ≡ √ e−u du. (5.16) π x

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 inputs. 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 interferometer 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: 1 P (1) = [3 + 2 cos(ka)+2cos(kb)+2cos(k (a + b))] (5.17) 12 81 × [3 + 2 cos(k a)+2cos(k b)+2cos(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: 1 P (2) = [9 + 2 cos(ka+ k a)+2cos(kb+ k b) (5.18) 12 81 +2cos(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: 1 P (3) = [3 + 2 cos(ka+ k a)+2cos(kb+ k b) (5.19) 12 81 +2cos(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 typical 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 interferometers 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: 1 1 1 1 P = P = + + = . (5.20) 1 2 9 9 9 3 1 The two-photon interferences of eq. (5.19) have a maximum of value of 9 . 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 (P ) 1 12 = . (5.21) P1 3 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 interferometers used in the two-photon three-path interference experiment. The six ﬁber-pigtailed detectors are visible in the center.

◦ 450 mW. A LiIO3 crystal (30 × 30 × 30mm)cutat90 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 wavelength and k-modes of the photon pairs:

hω¯ p =¯hω1 +¯hω2, (5.22)

h¯kp =¯hk1 +¯hk2. (5.23)

For our pump wavelength the photon pairs with center wavelengths of 632.8nm 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 adjustments could thus be made with laser light instead of tenuous downconversion light. Our single-photon counters use Silicon avalanche photodiodes that are very 72

polarization control phase control

APD detectors

l Det 4 m Det 5 filter s 788.7 nm fused fiber fused fiber coupler coupler Det 6 M4x Figure 5.3: Schematic of the 789 nm interferometer: A microscope objective precisely selects a k-direction corresponding to a wavelength selectivity of ∆λ =1.2nm. Because the coherence length determined by the mode selection and filters (lc = 250 µm) is much smaller than the path lengths differences (∆ > 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 setupbefore 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

1,0

0,9 1 nm

0,8

0,7

0,6 2 nm 0,5 λ = 633 nm 1 Two photon visibility λ = 789 nm 0,4 2 3 nm 0,3 0,0 0,5 1,0 1,5 2,0 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, 789 nm) and interference filters with three different bandwidths ∆λFWHM. Even for small bandwidths ≈ 1nm as in our experiment the reduction in visibility due to group velocity dispersion for path-length differences 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 (< 5 µm) 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 dispersion (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 dispersion 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: 2 2 2 1 λ 2 1 λ D = σ − D x + σ − D x . (5.24) 2 2πc λ 2 2πc λ

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 groupvelocity 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 microscope 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 200 µm a mechanical method of path-length adjustment had to be introduced.

phase control adjustable air gap APD detectors

l' adjustable air gap Det 1 m' Det 2 filter s' polarization 633 nm fused fiber control fused fiber Det 3 coupler coupler M4x Figure 5.5: Schematic of the 633 nm interferometer: The length of the air gaps can be adjusted using motorized translation stages (cf. photograph on p. 76).

Air gaps in the second interferometers (operating at 633 nm) allowed adjustment 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 objectives. 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 groupvelocity 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)inthe633 nm interferometer. The length of the air 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 interferometers 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 of volts produce a phase shift of several periods. 79 losses, they proved very practical for purposes of adjustments: a reverse illumination scheme could be used.

5.3.2 Polarization and path length adjustment

polarization controllers

l m

Si-PD M4x s M4x laser analyzer FC/PC connector polarizer Figure 5.9: The reverse illumination scheme for interferometric polarization adjustment uses a polarized laser connected to one detector output. The manual polarization controllers induce an optical retardation with a changeable optical axis. By tilting the disks the polarization state in the ﬁber is changed. The polarization state at the tritters is the same when the photodiode behind the analyzer (at 90◦ to the polarization of the downconversion 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 interferometers 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 black box. The three manual polarization controllers with tiltable disks were used to set the polarization in the interferometer. The phase shifters are in the white boxes in the foreground. The integrated ﬁber tritters are barely visible. 81 through a linear polarizer and coupled into the interferometers through the detectors 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 interferometers. 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,andm 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 coherence 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 illumination 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 photodiode behind the analyzer is zero. The signal at the photodiode was monitored as the disks of the manual polarization controllers were tilted. By iteratively reducing 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

m a) m' ∆x

λ s s'

b) m m' ∆x

s λ1 λ2 s' S

Figure 5.11: Interferometric path-length adjustment using both fiber interferometers: the long paths are not shown here. Fig. a) shows the reverse illumination scheme described in the text. Fig. b) shows the two-photon interference setup. If λ1 = λ2 = λ the path-length differences in the interferometers b) are exactly the same when white-light fringes are observed for case a). If the downconversion wavelengths are different this method will 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 gaplengths that is required to fulﬁll the white light interference condition 83

Scan over m+m' and s+s' fringes

300

250

200

150

100 Coincidences in 10s

1000 1500 2000 2500 3000 Position [µm]

Figure 5.12: Two-path interferences for (m + m and s + s) paths as the length 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

Scan over l+l' and s+s' fringes

300

250

200

150 Coincidences in 10s 100

2000 2500 3000 3500 Position [µm]

Figure 5.13: Two-path interferences for (l + l and s + s) paths as the length of the l air gap is changed.

5.3.3 Detection system

The ﬁber pigtailed Silicon avalanche photodiodes were cooled to about −30◦ C and biased in reverse to about 20 V above the breakdown voltage. Every incident photon releases an avalanche of carriers in the semiconductor detector. This macroscopic pulse is ampliﬁed, shaped and normalized by a constant fraction discriminator. 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

D DETECTION > CFD start

TAC PHA S stop D delay > CFD line

PZ PZ > counters DA > IEEE bus Plug-in M M

motor RS232 control shutter PC control LPT CONTROL Figure 5.14: Detection and control electronics. The detection system amplifies the signal from the detectors (D), shapes the pulses using the constant fraction discriminator (CFD). The pulses are registered in the counters connected to the IEEE-488 bus of the computer (PC) and used to start and stop a time-to-amplitude converter (TAC). The single count rates and the coincidence counts in the narrow time window set by the pulse height analyzer (PHA) are registered on the PC. The 633 nm interferometer is also computer controlled. The path lengths can be varied using two micro-positioning motors (M) with a nominal resolution of 0.1 µm controlled over the serial interface port. The phases are set by piezo tubes (PZ) driven by the amplified signals from a digital-to-analog converter (DA) which is controlled over the IEEE-488 bus. Finally, the electromagnetic shutter (S) used to switch the calibration laser on and off is 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

ss', mm', ll' 300

200 ms' ml' sl' sm' lm' 100 ls' Coincidences in 100s

-8 -6 -4 -2 0 2 4 6 8 τ [ns]

Figure 5.15: The histogram of time delay between detection events was recorded on the computer using the TAC and the PHA. The horizontal axis gives the time delay between detection events at one detector of the 789 nm interferometer and one detector of the 633 nm interferometer. The vertical axis gives the counts in a bin of ≈ 0.1ns. The rate at a given delay interval (coincidence window) is proportional to the surface under the curve. Two-photon interferences with maximal visibility can be observed when the coincidence window encloses only the central peak, which has the contributions from the s − s, m − m,andl − 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 possible 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,them-m, and the s-s paths is exactly the same, if we have properly adjusted the interferometers. 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 successively 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 aphotonpairtobecounted(Cxy) for a given path combination (xy)ispropor- tional to the product of the probabilities (Tx and Ty) that the single photons have transversed a given path. The proportionality parameter C 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 = CTlTs (5.27)

Cms + Clm = CTmTs + CTlTm (5.28)

Cml + Csm = CTmTl + CTsTm (5.29)

Csl = CTsTl (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 ,C can be solved. The following table summarizes the normalized transmissions for the paths and the coeﬃcient C as measured by this method:

633 nm 789 nm

Detectors 2 & 5 Ts Tm Tl Ts Tm Tl C 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 200 µm coherence length of the single photons the three contributions cannot, not even in principle, be distinguished.

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:

R633 R789 τc = 15200 cps 8300 cps 3.5ns=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 purpose a stable HeNe laser was coupled into a ﬁber beam splitter whose outputs were connected to the free inputs of the disbalanced Mach-Zehnder interferometers. The HeNe laser was strongly attenuated so that the single photon counting detectors and electronics could be used to monitor the outputs of the interferometers. 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 20 ms and closed in about 55 ms. The parallel printer port of the computer was 89

788.7 nm APD detectors from crystal l Det 4 m s Det 5

Det 6 attenuator

HeNe laser to other interferometer shutter control

Figure 5.16: Calibration of phase settings. The downconversion mode from the crystal is coupled into one input of the interferometer. Light from a strongly attenuated HeNe laser is coupled into the input of a ﬁber beam splitter. The outputs of the beam splitter are used as inputs to both three- path interferometers. A computer-controlled shutter switches the HeNe laser used for measurement of the single-photon calibration interferences on 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 step1.

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 upto 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 includes 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 stoppulse 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

ss', ms', lm'

300

ml' sl' 200 ms' sm'

ls' lm'

Counts in 100 s 100

2&5 3&5 1&5

Figure 5.17: Coincidences between one detector on one side and three on the other. The x-axis is the time delay between a count event at output detector 5 of the 789 nm interferometer and a count at one of the outputs of the 633 nm interferometer. By introducing an electronic time delay between these three counters all three coincidence rates could be monitored by the same electronics. Three time windows (shaded gray in the plot) were set on the interfering peaks. The integrated counts in these peaks were 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 −6 ◦ silica is on the order of kth =7×10 per C. For our interferometers this implies a phase shift of about 17 wavelengths per ◦ C and meter of fiber!

The ﬁber interferometers were enclosed in particle board boxes and insulated 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. 633

cal. 789

[°C] laboratory temperature 21.5 21.0 temp. in boxes 20.5 0612 Time [h]

Figure 5.18: Temperature drifts. Temperature in laboratory and in insulated interferometer boxes as function of time (lower traces). The other traces show the simultaneously recorded calibration fringes and coincidence counts. The temperature in the laboratory varied about 1◦ C during the 12 hours. The temperature in the boxes changed about 0.2◦ C in this time, with no perceptible ripple. The calibration fringes show a considerably 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 interferometers give an average fringe drift of 58 fringes/◦ C in each interferometer. Even when the temperature changed very slowly the interference fringes showed instabilities (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

2&5 80 3&5 1&5

Coincidences in 2s 20

0 20 30 40 50 60 70 Time [m]

Figure 5.19: Typical measurement showing three pairs of coincidences (1&5, 2&5, 3&5). Smoothed traces with high frequency noise removed are shown. The phases were changed by temperature variation in the ﬁber interferometers. One can recognize the 120◦ phase shift between the three pairs 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 4 1 4 1 4

2 5 2 5 2 5

3 6 3 6 3 6 0 2π/3 4π/3 Figure 5.20: Three possible pairs of correlations observed between detectors (1&5, 2&5, 3&5 as ﬁlled dots). The phase shift between the three possible 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 0.03941 Hz for the 633 nm interferometer and 0.01875 Hz for the other interferometer. We expect the maximum of the two-photon interference signal to be at: 633 nm 0.0544 Hz = 0.03941 Hz + 0.01875 Hz (5.32) 789 nm 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 0.054 Hz, 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

Coincidences [cps] 20

0 4 2 0 0 500 1000 1500 Piezo control [V] Time [s] Figure 5.21: Typical raw measurement showing high visibility interferences. One phase shifter in each of the two interferometers was driven by a saw-tooth signal voltage. The other phase drifted. The uncalibrated raw-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 + ω2)t + φ0]] . (5.35) Again one has to correct for the fact that the calibration laser has a diﬀerent wavelength: 633 nm ω = ω . (5.36) 2 2 789 nm

Only the parameters N0, V ,andφ0 were ﬁtted. The periodicity of the two-photon interferences was taken from the calibration fringes. 97

30 Calibration 789nm interferometer

25 0.03941 Calibration 633nm interferometer 0.01875 20

10 0.05371 Coincidences [633nm & 789nm]

0 ln ( maximum entropy spectral estimation) 0.00 0.05 0.10 0.15 f [Hz]

Figure 5.22: Dominant frequencies of the calibration (three traces for the 633 nm interferometer, two traces for the other interferometer) and two-photon interference fringes (lower dashed line). The calibration laser frequencies was 633 nm for both interferometers. The PDC coincidences were recorded at the wavelengths 633 nm and 789 nm. The quickly varying piezo voltage in one arm of each interferometer induces a periodic signal in both calibration (0.0186 Hz, 0.0394 Hz) and two-photon interference signal (0.0537 Hz). The slowly varying signals near zero frequency are 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

0π 1π 2π 3π 4π [rad]

Det. 1 Det. 2 Det. 3 Counts

1.0 1.5 2.0 2.5 3.0 3.5 [V] -0.5π 0.0π 0.5π 1.0π 1.5π [rad]

Counts Det. 4 Det. 5

1.0 1.5 2.0 2.5 3.0 3.5 [V] Control voltage

Figure 5.23: Typical calibration measurement. The measurement for the 633 nm interferometer is shown in the upper plot; the measurement for the 789 nm interferometer is shown in the lower plot. The phases calculated from the HeNe laser interferences which can be used to calibrate the phase shifts induced by the piezo and the drifts. For short times the phases in two arms is nearly constant and the phase in the arm modulated by the piezo varies linearly. The fringe pattern can then be well approximated by a sinusoidal. 99

π 6 ϕ + ϕ 5π m m' 3π ϕ 2π m' 0π ϕ -2π m

Phase shift [rad] 0204060Time [s]

20 Coincidences [Cps]

-2π 0π 2π 3π 5π 6π ϕ +ϕ [rad] m m'

Figure 5.24: Two-photon interference fringes. The phases were calibrated by using the HeNe fringes (see Fig. 5.23). One pair of relative phases was nearly constant during this measurement. The two-photon interference pattern can be well ﬁtted by a sinusoidal which is a function of the sum of the two changing phases. Because small drifts in the long or medium phases were not considered in the model the period of the best ﬁt curve is 0.95 × 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 measurement 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 parameters). These parameters can be calculated from the time histograms previously measured (cf. p. 87). The model function for the interferences is then:

2 2 2 2 2 2 Nc = N0 tsts + tmtl + tl tm (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 }.Wehavealready 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 } can be 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 coincidence 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 calculated 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 − V1V3 − V2V3 2V3 2V2 2V1

The absolute minimum (4) is only achievable if V V V V V V V V 2 3 − 2 3 ≤ 1 3 − 1 3 ≤ + 2 1and + 2 1, (5.41) 2V3 2V2 2V1 2V3 2V1 2V2 which can be assumed for our interferometer.

We estimated the visibility parameters for our experiment to be: I0 ≈

30.5cps,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

80 70 60 50 40 30 20 Coincidences [cps] 10 0 1000 1100 1200 1300 Time [s]

Figure 5.25: Section of the raw data (dots) with ﬁt curve (solid line). The ﬁt curve approximates the measured curves with the linear approximation to phase drifts as described in the text. The non-sinusoidal fringes observed 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 (Ω U(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 interpolation 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 coincidence 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

20 π

Coincidences [cps] Coincidences 10 1.5 1.0π 0.0π 0.5π 0.5π ' 1.0π ϕ 1 ϕ 1.5π 0.0π + + 2 ϕ' ϕ 1 2

Figure 5.26: Calibrated measured data averaged over bins of equal phase × settings. A 6 6 grid was used in the φ1 + φ1 and φ2 + φ2 parameter space. 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 measurements 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 experimentally 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 imperfect correlations by using the concept of fuzzy sets. The result would then be fuzzy elements of reality, or elements of reality having (possibly) high probabilities 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)mustallbelargerthan 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 interferences. 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 stability 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 interferometer 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 interferometer. 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! l l’

m click! click! m’

click! click! s s’

t x

D S D’

Figure 5.27: One classical picture assigns an element of reality to the emission of a pulse from the crystal. The space-time diagram has been simpliﬁed by projection into one space dimension. The pulses are detected at three diﬀerent times (depending on which path they have taken). This picture is wrong because it cannot explain the interferences observed in the 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

click! click!

s s’

m m’

l l’ t x

D S D’

Figure 5.28: Quantum picture of the two-photon three-path interference experiment. The space-time diagram is simplified by leaving out one space dimension. The emission time of a photon pair is not well defined. The only element of reality we have is the click at a detector. Different emission times are in principle indistinguishable and will therefore show interference as was observed in the experiment. can, if we wish, draw lines back in time and must then conclude that the photons registered must have been emitted from the crystal at different times. Because the CW pump laser has a long coherence time and spontaneous downconversion events occurring in this time are indistinguishable, the probability amplitudes for these events interfere.

The fact that we see two-photon interferences shows that it is impossible to assign elements of physical reality to the emission of a photon pair from the 109 crystal. The two-photon three-path interference experiment again manifests the intricate link between the information the observing subject can in principle have about the observed objects and the actions of these objects themselves. Experiments in quantum physics imply a concept of information fundamentally incomprehensible in a classical world. 110 Chapter 6

Conclusions and Outlook

In this work we have explored the quantum interferences in optical multiports. Our research has concentrated on the study of photon pairs from a parametric downconversion crystals in optical ﬁber multiports in the framework of the physics of entanglement and Bell’s inequalities. We have derived a new a Bell inequality for a correlated three-state system.

The theory of multiports has given an algorithmic proof that any discrete ﬁnite-dimensional unitary operator can be constructed in the laboratory using optical devices. Our recursive algorithm factorizes any N × N unitary matrix into a sequence of two-dimensional transformations. The experiment is built from the corresponding beam splitter devices. This also permits the measurement of the observable corresponding to any discrete Hermitian matrix and shows that optical experiments with any type of radiation (photons, atoms, etc.) exploring higher-dimensional discrete quantum systems can be simulated with multiport interferometers.

In the experimental part we have first characterized the fiber optical multiports (tritters) by building a three-path Mach-Zehnder interferometer. This interferometer shows the enhancement of sensitivity typical for multipath interferometry. Then we have demonstrated the operation of fiber optical multiports on individual quanta by measuring photon bunching at the outputs of the integrated 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 entangled 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 amplitudes from different emission times at one crystal interfere. These amplitudes are separated by the first tritter and reunited by the second tritter in each interferometer. 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 radiation 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 − scontributions. 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

D1 T1

l D3 m s

cw-pump s' m' D' l' 1 D'2

T2 D'3 Figure 6.1: New scheme for multipath time-energy entanglement. Three PDC crystals are pumped by a UV laser. Two modes from each crystal are coupled into the three inputs of two tritters. Six detectors monitor coincidences. There are no special requirements on the coherence length of the pump laser. The space-time diagram of this experiment is the same as the simplified version presented for our experiment (Fig. 5.28). all fibers can be made equally long. Finally, the phase modulation can be performed before the first tritter and thus the fiber part of the interferometer can be phase-stabilized by immersion in a stable temperature bath. 114 Appendix A

Devices

A.1 Parametric downconversion crystals

A.1.1 KDP and LiIO3

Figure A.1: Energy conservation and phase matching in the crystal in the parametric downconversion process determine the propagation direction and color of the emitted light. For type-I downconversion the cones for diﬀerent colors are coaxial. The polarization of downconversion photon pairs is the same.

◦ A Potassium Dihydrogen Phosphate (KH2PO4 or KDP) crystal (cut 90 to 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 experiments 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 downconversion photons are emitted on one cone of ordinary and one cone of extraordinary polarization. Along intersections of cones of equal wavelength polarization-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.Thecrystalwas 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 polarization. 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 Tritter 2’ (633 nm) no. F3766 In \ Out RB W In \ Out RB W R 32 35 33 R 35 31 34 B 39 29 32 B 32 37 31 W 32 38 30 W 33 32 35

Tritter 1 (780 nm) no. F3771 Tritter 2 (780 nm) no. G3374 In \ Out RB W In \ Out RB W R 38 30 32 R 33 32 35 B 36 37 27 B 34 34 32 W 28 35 37 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 = 351.1nmusingan internal etalon. The maximal power was approximately 1 W. The typical power used in the experiments was 450 mW. Further parameters of the pump laser are summarized in the following table: 120

Innova 400 Argon-Ion laser

Pmax 1000 mW

Ptyp 450 mW 1/e2 beam diameter 1.8mm divergence (full-angle) 0.4mrad coherence length > 2m

A.3.2 HeNe laser

HeNe Laser (Melles Griot 15mW, S/N 72058J) 0.5 439 MHz 0.4

0.3

0.2 Intensity [a.u.]

0.1

0.0 -1.0 -0.5 0.0 0.5 1.0 GHz

Figure A.3: Measured output spectrum of the HeNe laser. The mode separation of 439 MHz corresponds to a coherence length of 35 cm.Thefre- 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 15 mW. The mode separation of 439 MHz 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 10 mW

Ith 38.2mA Iop 63.9mA

λp 783 nm η 0.39 mW/mA

Im 3.07 mA ◦ ◦ 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 50 µm. 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 −30◦ C 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 390 ΚΩ h ν

D signal

50 Ω

Figure A.4: Passive quenching circuit used for the ﬁber-pigtailed silicon avalanche photodiodes. The diodes were reversely biased and operated 20 V 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

+12V

1N914 12V Solenoid 7404 TTL 1 ΚΩ

3.3 ΚΩ 1 ΚΩ

Figure A.5: TTL interface for the electromagnetic shutter. The line printer port of the computer was used as simple TTL output. The interface drives the 12 V solenoid which opens and closes the shutter (based on 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 voidlpt1_select_lo(void){ /* pointer to LPT1 IO Addr */ unsignedint far *plpt1=MK_FP(0x0040,0x8); unsignedchar lpt_control; /* control port select printer bit */ lpt_control|= 0x08; outportb(*plpt1+2,lpt_control); } voidlpt1_select_hi(void){ /* pointer to LPT1 IO Addr */ 125

unsignedint far *plpt1=MK_FP(0x0040,0x8); unsignedchar 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 −25◦ C

T [◦ C] = 2.5588 (R − 100) (A.1) with R the resistance [Ω]. 126 Appendix B

Reprint from Phys. Rev. Lett. 73, 58–61 (1994)

REPRINT

M. Reck, A. Zeilinger, H.J. Bernstein und P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).

127 128

PRL 129

PRL 130

PRL 131

PRL 132 Appendix C

A computer program for the design of unitary interferometers

C.1 Mathematica notebook

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 p Sin[w] -(E Cos[w])

I p Cos[w] E Sin[w]

133 134

Ye olde beam splitter Notice that the matrix need not be normalized. DrawExperiment[MatrixToExperiment[{{1,1},{1,-1}}]] 2

T=1/2 π 1

0 0

2 1 -Graphics- The Tritter MatrixForm[ 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 parameters {t,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 Pi {0.304087 Pi, ------} {------, ------} 3 4 3 0 135

Picture of experimental setup DrawExperiment[{t,a}] 3

(4*Pi)/3

T=1/2 π 2

(4*Pi)/3 (5*Pi)/6

T=2/3 T=1/2 π π 1

2*Pi 2*Pi 2*Pi

3 2 1 -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 quarter DrawExperiment[MatrixToExperiment[ DiagonalMatrix[{1,1,1,Exp[I*Pi/6]}].quarter //. p->Pi/3 ]]; 4

πT=1/2 3

Pi/3 Pi

πT=2/3 πT=0 2

Pi Pi Pi

πT=3/4 πT=2/3 πT=1/2 1

Pi/6 0 0 0

4 3 2 1 136 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

(* Parameters solving equation 0==M21*B11+M22*B21 for given beam splitter matrix. *) bsparam[m21_,m22_]:= Module[{w,p}, If[Chop[m21]==0, w= N[Pi/2]; p= 0;, (* skip *) If[Chop[m22]==0, w= 0; p= N[Pi];, (* swap beams (==mirror) *) p= Pi+Arg[m22]-Arg[m21]; (* transform *) w= ArcTan[Abs[m21],Abs[m22]]; ]; (* If *) ]; (* If *) Return[{w,p}]; 138

]; (* Module *)

(* Definition of error messages *) UnitaryMultiports::NonNumericError = " ‘1‘ not a numeric quantity." UnitaryMultiports::TransformationError = " in row ‘1‘ column ‘2‘ transformation of ‘3‘." UnitaryMultiports::UnitaryError = " matrix not unitary !"

(* Definition of exportable functions *) MatrixToExperiment[m_]:= Module[ {mx,mx0,dx,zx,n,r,c,eps,p,w,n2pi,alpha,t,i,j}, Clear[i,j]; mx0= N[m]; n= Dimensions[m][[1]]; (* normalize *) dx= Transpose[Conjugate[mx0]].mx0; zx= DiagonalMatrix[ Table[1/Sqrt[dx[[i,i]]],{i,1,n}] ]; mx= N[mx0.zx]; If[Chop[Transpose[Conjugate[mx]].mx] != IdentityMatrix[n], Message[UnitaryMultiports::UnitaryError]]; n2pi= N[2*Pi]; t=Table[0,{i,1,n},{j,1,n}]; (* empty array for result *) If[!MatrixQ[mx,NumberQ],Message[UnitaryMultiports::NonNumericError,mx], For[r=n,r>1,r--, (* all rows >1 *) For[c=r-1,c>=1,c--, (* all cols >= 1 *) (* calculate phase shift and reflectivity parameters *) {w,p}= Mod[ bsparam[ N[mx[[r,c]]],N[mx[[r,r]]] ], n2pi]; (* now multiply matrices *) mx= mx.bs[n,r,c,w,p]; (* test *) If[Chop[mx[[r,c]]]!=0, Message[UnitaryMultiports::TransformationError,r,c,mx[[r,c]]]; ]; (* insert in result *) t[[r,c]]={w,p}; ]; (* For c *) ]; (* For r *) (* Now calculate phases *) alpha= Chop[N[Table[Mod[Arg[mx[[i,i]]],n2pi], {i,1,n}]]]; ]; (* If *) Return[Rationalize[N[{t,alpha}/Pi]]*Pi]; ]; (* Module *)

(* the following functions are used to draw the setup *) bscube[x_,y_,dw_,txt_]:=Module[{t,c,d,t2}, t=Text[FontForm[StringForm["T=‘1‘",InputForm[txt]] ,{"Helvetica",8}],{x+dw,y+dw},{-1,-1}]; t2=Text[FontForm["p",{"Symbol",8}],{x+0.5*dw,y+0.5*dw},{-1,-1}]; c2=Circle[{x, y}, 0.5*dw, {0,Pi/2}]; 139

c=Line[{{x-dw,y+dw},{x+dw,y+dw},{x+dw,y-dw}, {x-dw,y-dw},{x-dw,y+dw}}]; d=Line[{{x-dw,y+dw},{x+dw,y-dw}}]; Return[ If[Chop[N[txt]]==1,t, If[Chop[N[txt]]==0,{t,c2,t2,d},{t,c2,t2,d,c}] ] ] ] phaseshifter[x_,y_,dw_,dp_,txt_]:= {Rectangle[{x-dw,y-dp},{x+dw,y+dp}], Text[FontForm[InputForm[txt],{"Helvetica",8}],{x+dw,y+dp},{-1,-1}]}

DrawExperiment[x_]:= Module[{t,a,dw,dp,n,i,j}, {t,a}=x; dw= 0.2; (* width of beam splitter cube *) dp= 0.05; (* height of phase shifter *) n= Dimensions[t[[1]]][[1]]; Show[Graphics[ {Table[ Table[ {bscube[i,n-j,dw,bstrans[t[[n+1-i,n+1-j,1]]]], phaseshifter[i,n-j+0.5,dw,dp,t[[n+1-i,n+1-j,2]]], Line[{{i-0.5,n-j},{i+0.5,n-j}}], (* H line *) Line[{{i,n-j-0.5},{i,n-j+0.5}}]} (* V line *) ,{i,1,j-1}] ,{j,2,n}], Table[ {Line[{{0,n-i},{0.5,n-i}}], (* H rays *) Text[n-i+1,{-0.05,n-i},{1,0}], Line[{{n-i+1,-0.5},{n-i+1,-1}}], (* V rays *) Line[{{n-i+0.95,-0.95},{n-i+1,-1},{n-i+1.05,-0.95}}], (* tips *) Text[i,{n-i+1,-1},{0,1}], Line[{{i-0.5,n-i},{i,n-i},{i,n-i-0.5}}]} (* corners *) ,{i,1,n}], Table[phaseshifter[i,-0.5,dw,dp,a[[n-i+1]]],{i,1,n}] } ],AspectRatio->1,PlotRange->All] ]

ExperimentToMatrix[l_]:= Print["In preparation !"];

End[ ] (* Private‘ *)

EndPackage[ ] (* UnitaryMultiports‘ *) 140 Appendix D

Wave packets in multiports

In this Appendix we will use the wave-packet formalism to ﬁrst describe the operation of multiports on a single photon and then on a two-photon state. Finally we will use it to derive the visibility function for a Mach-Zehnder interferometer with a dispersive element in one interferometer path. This section was motivated by a mistake found in [Rarity93] and later corrected in [Tapster94 ].

D.1 A single photon in a multiport

We can use the Heisenberg picture to describe the operation of a multiport interferometer on a single photon. The state impinging on the input port l of the ˆ† multiport is described by a wave-packet creation operator Al (f) with a single- † photon single-frequency creation operatora ˆl (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]. ∞ ˆ† | ≡ † | Al (f) vac dω fl(ω)âl (ω) vac (D.1) 0 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: ∞ √1 −iωt αl(t)= fl(ω)e dω. (D.2) 2π 0

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

∞ √1 −iωt αm(t)= Mml(ω) fl(ω)e dω (D.3) 2π l 0 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 :

T/2 ∗ Pm(T ) ∝ αm(t) αm(t) dt. (D.4) −T/2

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−ik(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: − 2 − (ω ω0) f(ω) ∝ e σ2 . (D.6)

The temporal wave-packet proﬁle after the device of length L then is

2 −iω (t−L/c) − σ (t−L/c)2 α(t) ∝ e 0 e 4 . (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]: ∞∞ ˆ † | ≡ † † | K (ζ) vac dω1 dω2 ζ(ω1,ω2)âl (ω1)âl (ω2) vac . (D.8) 0 0 The joint wave-packet profile is normalized: ∞∞ 2 |ζ(ω1,ω2)| dω1dω2 =1. (D.9) 0 0

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 correlation function: ∞∞ (2) ≡ Gmm (t1,t2) Mml(ω1)Mml (ω2)fl(ω1)fl (ω2) (D.10) l l 0 0 2 −i (ω1t1+ω2t2) × ζ(ω1,ω2) e dω1 dω2 . 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 (2) window of the detectors) is the integral over the Gmm (t1,t2) that describes the correlation at the outputs of the multiports: T/2 T/2 (2) ∝ Pmm (T ) Gmm (t1,t2) dt1 dt2. (D.11) −T/2 −T/2

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 Mml 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 or Mml 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 f

L 2 detector Figure D.1: Model system used to study the influence of material dispersion on the visibility of interference fringes. Light entering the interferometer has passed a Gaussian filter (f) and splits coherently at a symmetric beam splitter. One path contains a dispersive medium of length d; the other path is empty. The difference in length L = L1 − L2 is varied and 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) ∝ f(ω)e−iωt dω. (D.12) 0

The Mach-Zehnder interferometer can be described as a 2 × 2multiport with frequency dependent matrix elements. The matrix element taking the one 145 input to one output has two contributions: 1 eik(ω)L1 + eik(ω)(L2+d) . (D.13) 2

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 ∞ 1 Ψ(t) ∝ 1+eik(ω)(L+d) f(ω)e−iωt dω. (D.14) 2 0 The filter is assumed to be Gaussian

− 2 − (ω ω0) f(w) ∝ e 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 1 ∂2 k k(ω)=k(ω )+ (ω − ω )+ (ω − ω )2 (D.16) 0 ∂ω 0 2 ∂ω2 0 ω0 ω0

3 and discarding terms of O(ω − ω0) .

In the narrow bandwidth limit (ω0 σ) the integration can be extended to minus inﬁnity; the integral in equation D.14 thus evaluates to

2 − 2 σ2t2 − σ (t Tc) −iω t − −iω t + k (L + d) 1 − Ψ(t) ∝ e 0 e 4 + e 0 0 √ e 4 (1 iD) . (D.17) 1 − iD

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 frequency 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 groupvelocity vg, which in the case of dispersive media is not equal to the phase velocity vp of the wave:

ω0 vp(ω0)= , (D.18) k0 ∂ω v (ω )= . (D.19) g 0 ∂k ω0 146

The ﬁrst-order term in the Taylor expansion (D.16) describes the diﬀerence between groupand phase velocity.

The second-order term in the expansion 1 ∂2 k D ≡ σ2 d (D.20) 2 ∂ω2 ω0 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 : T/2 P (T ) ∝ |Ψ(t)|2 dt (D.21) −T/2 T/2 2 2 ∝ |G1| + |G2| +2|G1||G2| cos [∆ − δ] dt (D.22) −T/2 with two Gaussians σ2t2 G ≡ exp − (D.23) 1 4 1 σ2 (t − T )2 G ≡ √ exp − c (D.24) 2 1 − iD 4 (1 − iD) a path length dependent phase ∆ and a constant phase δ ω ∆ ≡ 0 (L + d) (D.25) c σ2 (t − T )2 δ ≡ c D. (D.26) 4 (1 + D2)

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 : 1 P = (1 + V cos(∆ − χ)) . (D.27) 2 147

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 overlapbetween the two interfering wave packets Tc and the dispersion in the medium D: 1 σ2T 2 V (σT ,D)=√ exp − c . (D.28) c 1 − iD 4(1 − iD)

Here i is the imaginary unit. The bars indicate the absolute value of the complex function. The visibility as function of σTc and D is shown in Fig. D.2.Evenwhen 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.

D 0

-2

-4

-3 -2 -1 0 1 2 3 σ Tc Figure D.2: Contours of equal visibility for a Mach-Zehnder interferometer with a dispersive medium in one arm. The visibility is lower than one even when the path lengths are optimally adjusted. The path length diﬀerence is proportional to σTc; the dispersion parameter D is deﬁned in equation.(D.20). The value of the function is indicated by the shading, which ranges from 0.0 (black) to 1.0 (white) in steps of 0.1. 148 Appendix E

Photographing Type-II downconversion

Photographs of downconversion light from a Beta Barium Borate (BBO) parametric downconversion crystal were take on high-speed infrared film using a 35 mm 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 (wavelength 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 photographs were developed for 4 min 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 165 mW. 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

1.0

0.8

0.6

0.4 UV.Sky UV.Haze Transmission 0.2 O2 Stack

0.0 300 400 500 600 700 800 λ [nm]

Figure E.1: Transmission curves of the cutoﬀ ﬁlters used to photograph type-II downconversion from BBO. 151

camera Iris high speed infrared film

BBO F1 IF2 F3 F4 F5 20mm

106mm

Figure E.2: Schematic of the setup used to photograph type-II downconversion: The BBO crystal is pumped by an Argon ion laser with P = 200 mW at 351 nm. An iris diaphragm helped to reduce background and reflected light. A tilted UV cutoff filter (UVSky F1) is used to reduce fluorescence from the exchangeable interference filter (IF2).We used 681 nm, 702 nm, and 725 nm interference filters with 5nm full-width-half-maximum bandwidth. A stack of cutoff filters (UVHaze F3, UVHaze F4, O2 F5) further reduce the background light. The camera is a normal 35 mm single-lens reflex camera with the lens removed. The typical exposure time for high speed infrared film was one hour. 152

Figure E.3: High speed infrared ﬁlm exposed with light from type-II downconversion in BBO. A 681 nm interference ﬁlter with 5nmbandwidth was used in a setup as shown in Fig. E.2.

Figure E.4: High speed infrared ﬁlm exposed with light from type-II downconversion in BBO. A 725 nm interference ﬁlter with 5nmbandwidth was used in a setup as shown in Fig. E.2. 153

Figure E.5: High speed infrared ﬁlm exposed with light from type-II downconversion in BBO. A 702 nm interference ﬁlter with 5nmbandwidth was used in a setup as shown in Fig. E.2. Polarization-entangled photons are observed at the intersection of the two circles. 154 Bibliography

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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, Jörg Schmiedmayer, Ernst M. Rasel, Markus Oberthaler, Raffael Egger, Roland Abfalterer, Johannes Denschlag, Sonja Franke, whose instruments 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 H¨opfel, 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

165 166

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 F¨orderung 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: Reck Vornamen: Michael Hunter Alexander Akademische Grade: Mag. Phil. und Mag. rer. nat. Geburtsdatum: 27. Juni 1961 Geburtsort: New York City, USA Familienstand: 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 Donauwörth, BRD. 10.1980–9.1981: Zivildienst in Garmisch-Partenkirchen und Germering. 1981-1987: Studium der Anglistik/Amerikanistik und Slawistik an der Universität Innsbruck (erste und zweite Diplomprüfung in Englisch und Russisch mit Auszeichnung) 27.6.1987: Sponsion zum Magister der Philosophie an der Universität Innsbruck 1987-1992: Studium der experimentellen und theoretischen Physik an der Uni- versität Innsbruck (erste und zweite Diplomprüfung mit Auszeichnung) 11.7.1992: Sponsion zum Magister der Naturwissenschaften an der Universität Innsbruck Seit 1992: Doktoratsstudium am Institut für Experimentalphysik der Universität Innsbruck

Auslandsstudienaufenthalte: 19.9.–30.9.1983: Slavistische Hochschulwochen der Universit¨at Konstanz, Deutschland (Intensivkurse Russisch und Serbokroatisch) 1.2.–1.3.1983: University of New Orleans, USA 1.9.–21.9.1984: Seminar f¨ur serbokroatische Sprache und Literatur in Belgrad, Novi Sad und Priˇstina, Jugoslawien 24.10.1985–24.01.1986: Puschkin Institut in Moskau, UdSSR (Intensivkurs Rus- sisch)

167 Berufliche Tätigkeit: 1987-1990 in den Semesterferien: Mitarbeit in der Forschung bei der Deutschen Luft- und Raumfahrt (DLR) in Oberpfaffenhofen, Deutschland 1988-1990: Lehrauftrag für Computerlinguistik an der Universität Innsbruck 1990 in den Semesterferien: Mitarbeit in der Forschung bei der Gesellschaft für Strahlenforschung (GSF), Neuherberg bei München Seit 1992: Vertragsassistent am Institut für Experimentalphysik, Universität Inns- bruck mit Forschungs- und Lehrtätigkeit (Ubungen¨ zur Mechanik und Wärme, Ubungen¨ zur Optik und verschiedene Laborpraktika)

Privat: Seit 1982 mit Renate Reck verheiratet. Unsere Tochter Anna Sophia wurde am 3. J¨anner 1991 geboren.

Publikationen

1. M. Reck und G. Schreier, “Fourier series representation of SAR polarimetric scattering signatures,” Proceedings of the 10th Annual International Geo- science and Remote Sensing Symposium, Washington DC, 5/1990.

2. M. Reck, G. M´endez und E. Gnaiger, “DatGraf - A graphical data analysis program. Application to non-invasive measurement of steady state ADP levels by stopped ﬂow mitochondrial respirometry,” Proceedings of the 5th Bio-Thermo-Kinetics Meeting, Bordeaux-Bombannes, France 9/1992.

3. M. Reck und A. Zeilinger, “Quantum phase tracing for the calculation of the transfer matrix for multiport-beamsplitters in experiments with correlated photons,” In Quantum Interferometry, hrsg. von F. DeMartini, G. Denardo und A. Zeilinger, Singapore, World Scientiﬁc, 1994.

4. M. Reck, A. Zeilinger, H.J. Bernstein und P. Bertani, “Experimental realization of any discrete unitary operator,” Phys. Rev. Lett. 73, 58 (1994).

5. M. Reck, A. Zeilinger, H. Bernstein und P. Bertani, “How to Build Any Discrete Unitary Operator in Your Laboratory”, Proc. of the 5th European Quantum Electronics Conference, Amsterdam, 43 (1994).

6. G. Weihs, M. Reck, H. Weinfurter und A. Zeilinger, “All-ﬁber three-path Mach-Zehnder interferometer,” Opt. Lett. 21 (4), 302 (1995).

7. G. Weihs, M. Reck, H. Weinfurter und A. Zeilinger, ,Two-photon interference in optical ﬁber multiports,” Phys. Rev. A 53, 863 (1996).

168