Applications and Characterisation of Correlations in Quantum Optics

Applications and characterisation of correlations in quantum optics

CHRISTIAN KOTHE

Doctoral Thesis in Microelectronics and Applied Physics KTH School of Information and Communication Technology Stockholm, Sweden, 2011 KTH Skolan för informations- TRITA-ICT/MAP AVH Report 2011:06 och kommunikationsteknik ISSN 1653-7610 Electrum 229 ISRN KTH/ICT-MAP/AVH-2011:06-SE SE-164 40 Kista ISBN 978-91-7415-979-0 Sweden

Akademisk avhandling som med tillstånd av Kungliga Tekniska Högskolan fram- lägges till oﬀentlig granskning för avläggande av teknologie doktorsexamen i mikro- elektronik och tillämpad fysik, tisdagen den 7 juni 2011 klockan 13:00 i sal FB52, AlbaNova Universitetscentrum, Kungliga Tekniska Högskolan, Roslagstullsbacken 21, Stockholm.

Tryck: Universitetsservice US AB iii

Abstract

Quantum optics offers a huge variety of exciting phenomena. Many of them are still in their infancy and especially when it comes to implementing devices using these effects for more than a proof of principle demonstration still many things have to be investigated and understood. In this thesis I discuss the role of correlations in some areas of quantum optics and in some cases compare it to classical optics. Four papers form the core of the thesis. In the first paper, I propose a new measure for entanglement. This measure is based on correlations between two states. I show, how this measure relates to another measure, the concurrence. It turns out that the measure is a bijective map of the concurrence for a pure state of two qubits. I motivate why the new measure is useful if one wants to implement it experimentally. I discuss its behaviour for the case of two qubits and show its properties when dealing with pure and with mixed states. The second paper extends the result of the first one to the case where one has higher-dimensional states than qubits. In the third paper I look at phase super-resolution. I show that it can be interpreted as a purely classical effect and I analyse what is needed and what is not needed to achieve it. Specifically, I show that quantum correlations in terms of entanglement is not needed to demonstrate phase super-resolution. By doing so I propose how one could achieve arbitrarily high phase super- resolution. Finally, the last paper looks at the efficiency of quantum lithography and quantum imaging. It shows, that some basic assumptions in the original proposals of quantum lithography seems unfounded and that, as a consequence, the efficiency is poor. I give formulæ for the explicit scaling behaviour when changing the number of photons in a mode or when changing the number of pixels. The effect of the results on the future of quantum lithography is discussed as well. iv

Sammanfattning

Kvantoptiken erbjuder ett stort antal spännande fenomen. Många av dem är fortfarande i sin linda och särskilt när man vill tillämpa kvantoptiska effek- ter snarare än att bara visa att principen fungerar så finns det många saker och ting som måste förstås och undersökas bättre. I denna avhandling ska jag diskutera vilken roll korrelationer spelar i några områden inom kvantoptik och i några fall ska jag jämföra dem med klassisk optik. Fyra vetenskapliga artiklar bildar kärnan i avhandlingen. I det första pappret föreslår jag ett nytt sammanflätningsmått. Detta mått har sin ursprung i korrelationer mellan två tillstånd. Jag visar hur måttet förhåller sig till ett annat mått, den så kallade ”concurrence”. Det visar sig att måttet är en bijektiv avbildning av concurrence för rena tillstånd av två qubitar. Jag motiverar varför det nya måttet är användbart när man vill implementera det experimentellt. Jag diskuterar hur måttet beter sig för två qubitar och visar dess egenskaper för rena och blandade tillstånd. Det andra pappret utvidgar första papprets resultat till situationer där man har tillstånd med högre dimension än qubitar. I det tredje pappret undersöker jag superfasupplösning. Jag visar att man kan tolka detta som en rent klassisk effekt och jag undersöker vad man be- höver och vad man inte behöver för att uppnå superfasupplösning. Jag visar särskilt att kvantkorrelationer genom sammanflätning inte behövs för att visa superfasupplösning. Därigenom ger jag förslag om hur man kan uppnå god- tyckligt hög superfasupplösning. Slutligen tittar jag i sista pappret på effektiviteten av kvantlitografi och kvantavbildning. Pappret visar att några grundläggande antaganden i origi- nalförslaget till kvantlitografi verkar vara illa underbyggda och att därigenom kvantlitografins effektivitet reduceras kraftigt. Jag ger ekvationer för det exak- ta skalningsbeteendet när man ändrar antalet fotoner i en mod eller när man ändrar antalet pixlar. Jag diskuterar också implikationerna som det medför för kvantlitografins framtid. Preface

This work consists of two parts. The first part is a summary of the work which I did during my PhD studies. Following that, I have attached four of my published papers. The work should be seen as a whole. Not everyhting in the attached papers is also discussed in the first part and not everything written in the first part is also part of any of my published papers. I rather make references in the first part to my papers when there is no need to repeat facts and thoughts. In most cases, however, the first part should be seen as an extension of the work written in the papers. It partly goes into more detail, explains concepts in another way, gives more introduction into the area, and links the papers together. The work presented in this thesis was performed under the supervision of Prof. Gunnar Björk in the Quantum Electronics and Quantum Optics group (QEO), which is part of the School of Information and Communication Technology at the Royal Institute of Technology (KTH) in Kista. My co-supervisor there was Prof. Anders Karlsson. The second half of my PhD-time I spent at the Quantum Infor- mation and Quantum Optics group (KIKO) at the Physics Department in Albanova at Stockholm’s university. During that time Prof. Mohamed Bourennane was the co-supervisor.

Acknowledgements

Hardly no work is the result of a single man and this thesis is by no means an exception. A lot of persons have had influence on it, let it be through discussions, through ideas, through critics or sometimes, only for the best, through preventing me from doing science. My first and highest thanks should go to my supervisors. Without the support, the guidance and, most importantly, the discussions with Gunnar Björk this work would not have been possible. For the second half of my PhD-studies the same should be said to Mohamed Bourennane, my co-supervisor during that time. Also Anders Karlsson, my co-supervisor during the first half of the PhS-studies receives my thanks. I had many discussions about science with other people as well. Without their willingness to answer my questions, to explain things or to give me useful hints many things would have taken longer times, would have been harder or maybe even impossible to perform. I want to thank from Kista especially Sébastien Sauge and Marcin Swillo for their great help in the lab and for explaining things to me, Maria Tengner, Anders Månsson and Piero G Luca Porta Mana for discussing physics with me and for help and cooperation when teaching, and Johan Waldebäck for help with some electronics. From AlbaNova I want to thank especially Hatim Azzouz for the help with the slits, Elias Amselem, Johan Ahrens and Magnus Rådmark for taking time to discuss things or to show me things in the lab every time I was wondering about something, Muhamad Sadiq for help in the lab and Per Nilsson for the help in programming and for the possibility to use his control software. I also want to thank Hoshang Heydari, Jonas Tidström, Mauritz Andersson, Daniel Ljunggren, and Hannes Hübel that you all provided time and answered questions when I went into your offices. A great thank should also be given to Eva Andersson for help with all administrative issues. Furthermore, I want to thank the above people and Aziza Sudirman, Jonas Almlöf, Atia, Alley, Amir, Aafke, Klaus, Philipp, Benjamin, Kate, Emma, Klas and Istvan for contributing to interesting lunch and “fika” diskussions and for having fun together at excursions. I enjoyed it very much. Not to forget José-Luis, Natasha and Isabel. I am very happy that we became friends and that we enjoyed a lot of things together. A special thanks goes also to Kårspexet and all its member. You were a very

vii viii ACKNOWLEDGEMENTS welcome diversion from science and a nearly constant source of joy and happiness. And thanks to Olaf, Aymeric and Anna, Lars, Frank, Dave and Kathrin for either skiing, going on holidays, or having gaming evenings together, for going out, or, most importantly, being around, being friends and having fun together. I want to thank Studienstiftung des deutschen Volkes for their support during my studies. Finally, I also thank my parents for their support. Without their help and motivation I would not have had the opportunity to defend my PhD thesis. The most special thanks, however, should go to Soﬁa and Malte. You are the most important persons in my life! Contents

Preface v

Acknowledgements vii

Contents ix

List of papers and contributions xiii Papers which are part of the thesis: ...... xiii My contributions to the papers: ...... xiv Paper which is not part of the thesis: ...... xiv Conference contributions: ...... xiv

List of acronyms and conventions xvii Acronyms ...... xvii Conventions ...... xviii

List of Figures xix

1 Introduction 1

2 Background in optics and quantum optics 5 2.1 Optics ...... 5 2.1.1 Polarisation ...... 5 2.1.2 Jones calculus ...... 7 2.1.3 Stokes parameters ...... 8 2.2 Quantum optics ...... 9 2.2.1 Qubit ...... 9 2.2.1.1 Several qubits ...... 9 2.2.1.2 Higher orders ...... 10 2.2.1.3 Bloch-sphere ...... 10 2.2.1.4 Implementation of the qubit ...... 13 2.2.2 Fock state ...... 13 2.2.3 N00N-states ...... 14

ix x CONTENTS

2.2.4 Coherent states ...... 14 2.3 Entanglement ...... 15 2.3.1 Concurrence ...... 16 2.3.2 Bell states ...... 17 2.4 Experimental components ...... 18 2.4.1 Lasers ...... 18 2.4.1.1 CW-lasers ...... 18 2.4.1.2 Pulsed lasers ...... 18 2.4.2 Nonlinear crystals ...... 18 2.4.2.1 Phase matching ...... 19 2.4.2.2 Spontaneous parametric down-conversion ...... 20 2.4.3 Optical components ...... 20 2.4.3.1 Beam splitter ...... 20 2.4.3.2 Polarising beam splitters ...... 21 2.4.3.3 Wave plates ...... 21 2.4.3.4 Polarisers ...... 22 2.4.3.5 Dichroic mirrors ...... 22 2.4.4 Detectors ...... 22

3 Phase resolution and sensitivity 25 3.1 Phase super-resolution ...... 25 3.1.1 Experiments with entanglement ...... 28 3.1.2 Experiments without entanglement ...... 29 3.1.3 Experiments without entanglement or joint detection . . . . . 29 3.1.3.1 Experiment in the space-domain ...... 31 3.1.3.2 Experiment in the time-domain ...... 33 3.1.3.3 Wavelength ...... 36 3.2 Phase super-sensitivity ...... 37 3.2.1 Simple approach ...... 37 3.2.2 Fisher-information ...... 38 3.2.3 Experimental results ...... 39 3.3 Summary ...... 39

4 Quantum imaging and lithography 41 4.1 Theoretical background ...... 41 4.1.1 The classical limit ...... 41 4.1.2 How to overcome the classical limit ...... 43 4.1.2.1 Using “classical” schemes ...... 43 4.1.2.2 Using “quantum” schemes ...... 43 4.2 Experiments ...... 44 4.2.1 Experiments with interferometers ...... 44 4.2.2 Experiments with slits and gratings ...... 44 4.2.3 Further experiments ...... 45 4.2.4 Known limits ...... 45 CONTENTS xi

4.3 Eﬃciency ...... 45 4.3.1 The theory of Boto et al...... 48 4.3.2 The theory of Steuernagel ...... 49 4.3.3 Comparison of the theories of Steuernagel and Boto et al. . 49 4.3.4 Conclusive experiment ...... 51 4.3.4.1 Idea of the experiment ...... 51 4.3.4.2 Realisation of the experiment by Peeters et al. . . 53 4.3.5 Theoretical analysis ...... 53 4.3.5.1 Analysis of the experiment of Peeters et al. . . . . 53 4.3.5.2 Analysis of the original experimental proposal by Boto et al...... 55 4.4 Implementation of the concluding experiment in practice ...... 55 4.4.1 Crystal ...... 55 4.4.2 Slit ...... 56 4.5 Outlook and open questions ...... 57 4.5.1 Are there any other possible correlations? ...... 58 4.5.2 Other investigations about linear scaling behaviours . . . . . 61 4.5.3 Using other media ...... 61 4.5.4 Outlook ...... 61

5 Measurement of entanglement 63 5.1 Introduction ...... 63 5.2 Quantifying entanglement through correlations of local observables . 64 5.2.1 Pure states of two qubits ...... 65 5.2.2 Mixed states of two qubits ...... 65 5.2.3 Higher-dimensional systems ...... 66 5.3 Outlook and open questions ...... 67

6 Summary and conclusion 69

A Background in mathematics 71 A.1 Linear algebra ...... 71 A.1.1 Vector space ...... 71 A.1.2 Hilbert space ...... 71

B Basics of quantum mechanics 73 B.1 The axioms of quantum mechanics ...... 73 B.2 Measurements ...... 74 B.3 Density matrices ...... 74 B.3.1 Pure states ...... 75 B.3.2 Mixed states ...... 75 B.3.3 Partial trace ...... 75

Bibliography 77

List of papers and contributions

Papers which are part of the thesis:

Paper A

Christian Kothe and Gunnar Björk, Entanglement quantiﬁcation through local observable correlations, Physical Review A 75, 012336 (2007), arXiv:quant-ph/0608041 (2006). Included in: Virtual Journal of Quantum Information, vol. 7, no. 2, 2007, and Virtual Journal of Nanoscale Science & Technology, vol. 15, no. 6, Feb. 12, 2007.

Paper B

Christian Kothe, Isabel Sainz, and Gunnar Björk, Detecting entanglement through correlations between local observables, Journal of Physics: Conference Series 84, 012010 (2007).

Paper C

Christian Kothe, Gunnar Björk, and Mohamed Bourennane, Arbitrarily high super-resolving phase measurements at telecommunication wavelengths, Physical Review A 81, 063836 (2010), arXiv:1004.3414.

Paper D

Christian Kothe, Gunnar Björk, Shuichiro Inoue, and Mohamed Bourennane, On the eﬃciency of quantum lithography, New Journal of Physics 13, 043028 (2011), arXiv:1006.2250.

xiii xiv LIST OF PAPERS AND CONTRIBUTIONS

My contributions to the papers:

Paper A I proposed G as a measure of entanglement, investigated its properties, did the simulations and wrote the paper.

Paper B I did the main contributions to the results, except for those presented in section 3 of the paper. I was not the main author.

Paper C I constructed the experiment, performed it, wrote some of the steering software, analysed the outcome of the experiment, did the calculations and wrote the paper.

Paper D I tried to perform a similar experiment as the one refered to in the paper, I did the calculations and the analysis, and I wrote the paper.

Paper which is not part of the thesis:

Isabel Sainz, Christian Kothe, and Gunnar Björk, Estimating entanglement through local correlations, Proceedings of SPIE 6726, 67263E (2007).

Conference contributions:

Hoshang Heydari, Christian Kothe, and Gunnar Björk, Quantiﬁcation of multipartite entanglement, invited talk 7.1.3 at the 14th International Laser Physics Workshop, Kyoto, Japan, July 4-8, 2005.

Christian Kothe, Simon Samuelsson, and Gunnar Björk, Improved ways of detecting entanglement, invited talk at the International Conference on Quantum Optics (ICQO 2006), Minsk, Belarus, May 26-31, 2006. CONFERENCE CONTRIBUTIONS: xv

Gunnar Björk, Christian Kothe, and Simon Samuelsson, Local uncertainty relations, covariance, and entanglement, invited talk 7.7.1 at 15th International Laser Physics Workshop, Lausanne, Switzerland, July 24-28, 2006.

Christian Kothe, Isabel Sainz Abascal, and Gunnar Björk, Detecting entanglement through correlations between local observables, invited talk at Quantum Optics III, Pucón, Chile, November 27-30, 2006.

Gunnar Björk, Isabel Sainz Abascal, and Christian Kothe, Estimating entanglement through local correlations, invited talk at the International Conference of Coherent and Non-linear Optics, Minsk, Belarus, May 28 - June 1, 2007.

Christian Kothe and Gunnar Björk, Quantum imaging and lithography, invited talk at 16th International Laser Physics Workshop, León, Mexico, August 20-24, 2007.

Christian Kothe, Gunnar Björk, and Mohamed Bourennane, Arbitrarily high super-resolving phase measurements without entanglement or joint detection, invited talk at 19th International Laser Physics Workshop, Foz do Iguaçu, Brazil, July 5-9, 2010.

Gunnar Björk, Christian Kothe, Mohamed Bourennane, and Shuichiro Inoue, Quantum lithography - possibilities and limitations, invited talk at Quantum Optics V, Cozumel, Mexico, November 15-19, 2010.

Gunnar Björk, Jonas Söderholm, Andrei B. Klimov, Luis L. Sánchez-Soto, and Christian Kothe, Polarization description of quantized ﬁelds, invited talk at the Max Planck Institute for the Science of Light Winter conference, Scheﬀau, Austria, February 28 - March 5, 2011.

Posters:

Christian Kothe and Gunnar Björk, Quantum lithography, contributed poster at Optik i Sverige 2007, Skellefteå, Sweden, November 8-9, 2007. xvi LIST OF PAPERS AND CONTRIBUTIONS

Christian Kothe, Gunnar Björk, and Mohamed Bourennane, A lot of fringes: super-resolving phase measurements without entanglement or joint detection, contributed poster at the International Conference on Quantum Information and Computation, Stockholm, Sweden, October 4-8, 2010.

Christian Kothe, Gunnar Björk, and Mohamed Bourennane, A lot of fringes: super-resolving phase measurements without entanglement or joint detection, contributed poster at Optikdagarna 2010, Lund, Sweden, October 19-20, 2010.

Gunnar Björk, Christian Kothe, Mohamed Bourennane, and Shuichiro Inoue, Limits to the eﬃciency of quantum lithography, contributed poster at CLEO/Europe-EQEC, München, Germany, May 22-26, 2011. List of acronyms and conventions

Acronyms

APD avalanche photodiode

BBO β-barium borate

BS beam splitter

CW continuous wave fs femtosecond

H horizontal

HWP half-wave plate iﬀ if and only if

L left

LO local oscillator

LOCC local operations and classical communications

MUB mutually unbiased basis

PBS polarising beam splitter

PNRD photon number-resolving detector

PP partial polariser

PS phaseshifter

QWP quarter-wave plate

R right

SMF single-mode ﬁbre

xvii xviii LIST OF ACRONYMS AND CONVENTIONS

SPD single-photon detector SPDC spontaneous parametric down-conversion SQL standard quantum limit Ti:S titanium-sapphire UV ultraviolet V vertical + plus - minus

Conventions

The following conventions are used throughout the thesis:

~a,~b, . . . a vector · scalar product A, B,... a matrix A,ˆ B,...ˆ an operator 1 matrix identity |φi , |ψi ,... vector in a Hilbert space hφ|ψi , hδ|γi ,... scalar product between vectors in a Hilbert space D E D E Aˆ , Bˆ ,... expectation value of an operator |0i , |1i ,... qubit |0i , |1i ,... state in the Fock space ≈ √approximately i −1 i, λ, x, β, . . . variables or indices O(...) order of e Euler’s number: 2.718 ... ∝ proportional to ⊗ tensor product (...)T transpose of a vector List of Figures

2.1 Optical polarisation ...... 6 2.2 Bloch-sphere of a qubit ...... 11 2.3 Bloch-sphere of a two-qubit state ...... 12 2.4 Schematic sketch of spontaneous parametric down-conversion and a beam splitter ...... 20

3.1 Mach-Zehnder interferometer ...... 26 3.2 Demonstration of phase super-resolution ...... 27 3.3 Illustration of Eq. (3.7) for n = 4 ...... 30 3.4 Splitting of coherent states ...... 31 3.5 Setup in space-domain ...... 32 3.6 Six-fold phase super-resolution (space-domain) ...... 33 3.7 Setup in time-domain ...... 34 3.8 Ten-fold phase super-resolution (time-domain) ...... 35 3.9 30-fold phase super-resolution (time-domain) ...... 36

4.1 Rayleigh limit ...... 42 4.2 Scheme interference experiment ...... 47 4.3 Detection scheme interference experiment ...... 51 4.4 Diﬀerence between Steuernagel’s model and the model of Boto et al. . 52 4.5 Diﬀerent theoretical plots of Eq. (4.15) ...... 54 4.6 Production of the double-slit ...... 58

5.1 Relation between G and C ...... 66

xix

Chapter 1

Introduction

Quantum mechanics is an “old” theory. Its foundations were laid by Max Planck more than one hundred years ago [Pla00]. Around thirty years later it seemed that by the formulation of a mathematical framework to calculate things, especially through Heisenberg, together with Born and Jordan [Hei25, BJ25, BHJ26], Schrödinger [Sch26c, Sch26e, Sch26b, Sch26d], and Dirac [Dir30], most things were discovered. Nothing could be more wrong. Already shortly afterwords Einstein, Podolsky and Rosen [EPR35] pointed out that quantum theory predicted counter- intuitive results which no one had thought of before. However, whereas they thought that quantum mechanics was “incomplete” and maybe should be replaced by something different, it turned out that their paper rather triggered a triumphal success1 of quantum mechanics. A first convincing answer to their arguments was given by Bohr, but the “hot stuff” started by the invention of the laser [Mai60, ST58], the foundation of quantum optics (among others by Glauber [Gla63a, Gla63b]), the Bell inequalities [Bel64] and finally their first violation by Aspect [AGR81]. Ironi- cally, Einstein himself, being the first to formulate the theory of stimulated emission [Ein16, Ein17], on which the laser principle is based, paved such the way to con- vince people that his doubts about quantum mechanics [EPR35] were unfounded and that quantum mechanics today stands stronger than ever before. Despite a lot of advances during recent times there are still a lot of things that remain unclear. There are two main points I want to discuss. One is the borderline between classical theory and quantum mechanics. It sometimes happens that new effects are discovered in quantum mechanics which subsequently turn out to be purely classical. One is the so called phase super-resolution, which I will discuss in section 3.1. Until some years ago it was treated as a sign of entanglement, and whereas the entanglement was dropped recently, I will show in this thesis that there is nothing quantum mechanical at all at phase super-resolution. Another effect is the Rayleigh limit which I will discuss in chapter 4. It was shown around 10 years

1According to some estimations around “30 percent of the U.S. gross national product is based on inventions made possible by quantum mechanics” [TW01].

1 2 CHAPTER 1. INTRODUCTION ago that this limit can be surpassed, in principle by an arbitrarily factor, with the help of quantum mechanics, a phenomenon called quantum lithography and quantum imaging. However, it turned out that also classical effects can beat the Rayleigh limit and as I will show in this thesis the efficiency of beating the Rayleigh limit in the quantum mechanical way is so low, that it may practically rule out any commercial usefulness of that method. The classical methods seem to be superior. However, this should by no means indicate that quantum mechanics is use- less. Usefulness in interferometers or other devices arises rather from phase super- sensitivity (see section 3.2) than from phase super-resolution and nothing indicates that this effect can be achieved without the help of quantum mechanics. Neverthe- less it is, in my opinion, important to make clear destinctions, both between the two concepts of phase super-resolution and phase super-sensitivity, and between which theory can be used to explain what. In the same way, although I show that quantum imaging seems to be unefficient, nothing points out that to overcome the Rayleigh limit with “classical” light can be done with an arbitrary small factor. Another question which is not fully examined yet, is the role of correlations in entanglement. Not all kinds of correlations allowed by quantum mechanics are possible in classical physics. Some of these “extra” correlations could give rise to the entanglement of states and I examine the relation between correlations and entanglement in chapter 5. Also the investigations about quantum lithography and quantum imaging as described above are mainly based on the question what kind of correlations photons can have. There, it turns out not so much in the kind one wants to have. Not to forget that it turns out that no correlations at all are needed for phase super-resolution. Meaningless to say: Every solved problem poses a plethora of new questions. This is the case in my thesis as well. Showing that phase super-resolution is purely classical does by no means answer the question how to achieve phase super- sensitivity in the best way. Showing that quantum lithography is not efficient in linear media (like air or vacuum) poses the question if there are other ways to keep photons staying together. And also the question about the connexion between correlations and entanglement is far away from being solved completely. This thesis is organised as follows: Those not familiar with the basic concepts of quantum mechanics may first want to read the appendices. For everyone else an introduction into the most important physical concepts and the most important experimental components I use is given in chapter 2. Those already familiar with it may want to skip this chapter. The next three chapters are devoted to the three main questions I spent my time on during the last years. First, in chapter 3, I will look into phase super-resolution and phase super-sensitivity, then, in chapter 4, I will discuss the efficiency of quantum lithography and quantum imaging setups. Af- ter that, in chapter 5, I will look at the question how one can make a quantitative measure of entanglement that is easy to implement experimentally. In chapter 6 I give a summary and conclusions of the whole work. Finally, after the appendices and the bibiliography, I have included four of my published papers, one for chapter 3, one for chapter 4 and two for chapter 5. 3

I wish you much pleasure in reading the thesis, or, in case you will not have it, that you at least will learn something new!

Chapter 2

Background in optics and quantum optics

This chapter should provide the reader with a suﬃcient background in physics (mainly optics and quantum optics) to understand the thesis. If someone is interested in these things in more details, or wants to get it in a broader context I can recommend the books of Born and Wolf [BW99] and Hecht [Hec03] for classical optics, the books of Schwabl [Sch02, Sch00] for quantum mechanics in general, as well as the books of Gerry and Knight [GK05] and Fox [Fox06] for quantum optics. For some deeper insights into quantum optics and the areas covered in this thesis I would suggest reading the book of Nielsen and Chuang [NC00] and the book of Mandel and Wolf [MW95].

2.1 Optics

2.1.1 Polarisation Light can be described as electromagnetic radiation at certain frequencies. There- fore, it is possible to use electromagentic wave theory when treating light mathematically. In classical optics one can under some conditions (looking at the far field and using homogeneous and isotropic media) define a wave vector ~k which points in the propagation direction of the light beam and two vectors E~ and B~ which are perpendicular to each other and to ~k and which point in the oscillation direction of the electric and the magnetic field, respectively. Light is called linearly polarised if the oscillation of the electric (or magnetic) wave-vector lies along a line, it is called circularly polarised if |E~ | (and |B~ |) is constant, but the direction of the vectors changes at a constant rate. Depending on the direction of the change one distinguishes between left-hand polarised light and right-hand polarised light. Light is called elliptically polarised if it can be described by a superposition of linearly and circularly polarised light. Fig. 2.1 shows a clarification. If E~ (and B~ ) change

5 6 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

Figure 2.1: Optical polarisation: If one assumes that ~k points in the z-direction (i.e., into or out of the paper) then E~ (or B~ ) is represented by the dotted line. The thick line illustrates the trajectory of E~ under different times and the arrows indicate in which direction E~ is moving. On the left linearly polarised light is illustrated, in the middle circularly polarised light and on the right elliptically polarised light. in a random manner light is called unpolarised. For a better quantification of polarisation one can define the coherence matrix

∗ ∗ Jxx Jxy hExExi ExEy J = = ∗ ∗ , (2.1) Jyx Jyy hEyExi EyEy ~ ~ where one assumes that k = |k|~ez and Ex and Ey are the components of E~ . Com- pletely polarised light is then deﬁned by |J| = 0, completely unpolarised light by the conditions Jxx = Jyy and Jxy = Jyx = 0, where both conditions are rotationally invariant. Linearly polarised light is represented by one of the matrices 1 0 0 0 J = I , J = I , (2.2) 0 0 0 1 or any rotation thereof, and circularly polarised light by 1 1 ±i J = I , (2.3) 2 ∓i 1 where I denotes the intensity of the light. Since any coherence matrix can be written as a sum of a completely polarised and a completely unpolarised coherence matrix one can deﬁne a quantity s 4|J| P = 1 − 2 (2.4) (Jxx + Jyy) as a degree of polarisation. This quantity is the ratio between the intensity of the polarised part and the total intensity and P is therefore bounded between 0 and 1. This degree of polarisation is invariant under any polarisation transformation [i.e., SU(2)]. 2.1. OPTICS 7

2.1.2 Jones calculus The Jones calculus was introduced by R. Clark Jones in 1941 [Jon41] and describes how polarised light evolves through birefringent media and gives one the possibility to treat the eﬀect of wave plates (see section 2.4.3.3), polarisers (see section 2.4.3.4) and other optical components in a simple way. Without loss of generality one can assume that ~k points in the z-direction. As a consequence E~ will only have components in the x- and y-direction and one can deﬁne a vector E ~ε = x , (2.5) Ey called the Jones vector. The Jones vector of an elliptically polarised beam will therefore be a cos ϕ − ib sin ϕ ~εL = E~ , (2.6) ell a sin ϕ + ib cos ϕ where a is the major axis and b the minor axis of the normalised ellipse and ϕ is the angle between the x-axis and the major axis of the ellipse. (If E~ changes its R L∗ orientation in the opposite direction the state would be described by ~εell = ~εell , where the asterix denotes complex conjugation.) Linearly and circularly polarised light can be seen as the degenerate cases of elliptically polarised light and are therefore included in the description of Eq. (2.6). A component in the optical system can than be described by a unitary 2 × 2 matrix M, the so called Jones matrix. After passing the optical component the state of the light can be described by a new vector

~εout = M~ε. (2.7) In such a way a beam after passing n optical components with their respective Jones matrices M1, M2,... Mn can be calculated as

~εout = MnMn−1 ... M1~ε. (2.8) The two most important components are probably retardation plates and polarisers. Retardation plates have the Jones matrices

e−iφ/2 0 M x = e−iΦ , (2.9) ret 0 eiφ/2 where 1 2π Φ = (n + n ) d (2.10) 2 s f λ and 2π φ = (n − n ) d (2.11) s f λ 8 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS are the mean phase change and the phase retardation, respectively, d is the thickness of the retardation plate, ns and nf are the indeces of refraction for the slow and the fast axis, respectively, and λ is the wavelength of the used light. In that case, one assumes that the plate is oriented so that the slow axis coincides with the x-axis and the fast axis with the y-axis. A linear polariser with the transmission axis in x-direction has the Jones matrix 1 0 M x = . (2.12) pol 0 0 If the objects are not oriented as assumed above they can be described by the Jones matrix M R = R(−θ)MR(θ) (2.13) instead, where cos θ sin θ R(θ) = (2.14) − sin θ cos θ is a rotation matrix and θ the rotation angle of the object with respect to the x-axis and with the z-axis as the rotation axis.

2.1.3 Stokes parameters The classical Stokes parameters were introduced by G. G. Stokes in 1852 [Sto52] and are deﬁned as 2 2 s0 = Ex + Ey , (2.15) 2 2 s1 = Ex − Ey , (2.16)

s2 = 2 hExEy cos δi , (2.17)

s3 = 2 hExEy sin δi , (2.18) where δ is the phase diﬀerence between Ex and Ey. They can also be expressed in terms of the coherence matrix of Eq. (2.1) as

s0 = Jxx + Jyy, (2.19)

s1 = Jxx − Jyy, (2.20)

s2 = Jxy + Jyx, (2.21)

s3 = i(Jyx − Jxy). (2.22)

The Stokes parameters are experimentally easily accessible; s0, for example is the total intensity. The degree of polarisation [Eq. (2.4)] can be expressed in terms of the Stokes parameters as ps2 + s2 + s2 P = 1 2 3 . (2.23) s0 2.2. QUANTUM OPTICS 9

2.2 Quantum optics

For those not familiar with quantum mechanics or the concept of density matrices a short introduction is given in Appendix B.

2.2.1 Qubit The basic ingredient in quantum information theory is the so called qubit. This word is a short expression for quantum bit. Whereas a classical bit (which itself is a short expression for binary digit) can consist of two distinct values (usually denoted 0 and 1), a qubit can either be in a state |0i or |1i, or any superposition of them, i.e., in general, a qubit |Qi can be written as

|Qi = α0 |0i + α1 |1i , (2.24)

1 2 2 where α{0,1} ∈ C and for the sake of normalisation |α0| + |α1| = 1. When doing a projection-measurement on one of the basis states |0i or |1i the probability of 2 2 ﬁnding |Qi in the state |0i (|1i) is then given by |α0| (|α1| ). The advantages of qubits arise from the fact that for operational purposes one might not have to do a projection measurement and therefore can apply an operation Uˆ on both basis states of the qubit simultaneously, whereas classical bits would require repeating a classical operation U on several input states.

2.2.1.1 Several qubits Just as one can combine single bits to chains of bits like 0110100..., one can combine several qubits |Q1i, |Q2i, ..., |Qni to form a new state

|Qi = |Q1i ⊗ |Q2i ⊗ ... ⊗ |Qni (2.25) n 1 X X (1)E (2)E (n)E = αi αi . . . αi ψ ⊗ ψ ⊗ ... ⊗ ψ , (2.26) 1 2 n i1 i2 in j=1 ij =0 E where each ψ(n) can have the value |0i or |1i. Often the tensor products are in omitted, so that the notation becomes

n 1 X X (1)E (2)E (n)E |Qi = αi αi . . . αi ψ ψ ... ψ (2.27) 1 2 n i1 i2 in j=1 ij =0 or, even more simply,

n 1 X X (1) (2) (n)E |Qi = αi ,i ,...,i ψ , ψ , . . . , ψ . (2.28) 1 2 n i1 i2 in j=1 ij =0 10 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

2.2.1.2 Higher orders Even if classical computers usually calculate in the binary system, one can also use any other basis like 3, 4 and so on. If one transfers this concept into the quantum mechanical case, it is called a qutrit, if one has a 3-level system and, in general, qudit or qunit if one has a d-level or an n-level system. In general, one can represent a d- level system in a Hilbert-space Hd−level = Cd. Here, one can take the tensor product d−level d−level⊗n as well if one has more than one d-level system Hn = H . The basis vectors of a d-level system can be named |0i , |1i ,... |di. Even combinations of diﬀerent-level systems are possible. The notation is straightforwardly extended to this case. One should also notice that some states are algebraically equivalent, for example is a two-qubit system algebraically equivalent to one four-level system. The most general pure state of n qudits can then be written as

n dj −1 X X (1)E (2)E (n)E |ψi = αi ,i ,··· ,i ψ ⊗ ψ ⊗ · · · ψ , (2.29) 1 2 n i1 i2 in j=1 ij =0 where each ψij is a dj-level system. Even here, one can use short-hand-notations similar to those in Eq. (2.27) and Eq. (2.28).

2.2.1.3 Bloch-sphere An often used representation of a two-level system like a qubit is the so called Bloch-sphere (see Fig. 2.2). In this representation the two poles correspond to the states |0i and |1i. By rewriting Eq. (2.24) as

|Qi = cos(ϑ/2) |0i + eiϕ sin(ϑ/2) |1i (2.30) with 0 ≤ ϑ ≤ π and 0 ≤ ϕ < 2π one can construct a unique one-to-one correspondence between a qubit state with defined values of ϑ and ϕ and a point on the surface of the Bloch sphere as shown in the figure. Two qubits are orthogonal if and only if (iff) the line connecting their corresponding points on the Bloch sphere passes the centre of the sphere. Any pure or mixed two-level system ρˆ can be uniquely written as 1 ρˆ = ~x · ~σˆ (2.31) 2 T ˆ T with ~x = (1, x1, x2, x3) and ~σ = (ˆσ0, σˆ1, σˆ2, σˆ3) , where σˆ0 = 1ˆ and the other entries are the Pauli matrices 0 1 0 −i 1 0 σˆ = , σˆ = , σˆ = , (2.32) 1 1 0 2 i 0 3 0 −1 even sometimes called σˆx, σˆy, σˆz, or simply X,Y,Z. Therefore, one can treat 3 (x1, x2, x3) ∈ R as a vector which represents ρˆ as a point on or - in case of 2.2. QUANTUM OPTICS 11

Figure 2.2: Bloch-sphere of a qubit with the basis states |0i and |1i. If one uses the representation |0i = |Hi and |1i = |Vi then the intersections of the axes with the sphere correspond to the states deﬁned in section 2.2.1.4. mixed states - inside the Bloch sphere. The maximally mixed state will then be at the centre of the sphere. The eigenstates of the Pauli matrices are placed on the respective axes. The Bloch sphere is an implementation of the Riemann sphere, which attributes every number c ∈ C in the complex plane and ∞ a unique number on the surface of the sphere. To see this, Eq. (2.30) can be rewritten as 1 |Qi = q (|0i + c |1i) , (2.33) 1 + |c|2 since the overall phase does not matter in quantum mechanics. With this technique any higher-dimensional state which is restricted and charac- terised by only one complex parameter (or equivalent by two angles as in Eq. (2.30)) can be illustrated as a point on a Bloch-sphere. As an example, one can use the general two qubit-state

|QQi = a00 |0i ⊗ |0i + a01 |0i ⊗ |1i + a10 |1i ⊗ |0i + a11 |1i ⊗ |1i (2.34) with the sum of the absolute square of the coeﬃcients equal to one and restrict it to the case where an exchange of the qubits will leave the state unchanged. With 12 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

Figure 2.3: Bloch-sphere of the exchange-invariant two-qubit state. The two-photon visibility Vcon = |sin(ϑ) cos(ϕ)| can be read out from the sphere as indicated (when taking the absolute value) and the concurrence (see section 2.3.1) can be calculated p 2 via C = 1 − Vcon. Therefore, all states lying in the interception of the sphere with a plane containing both the vertical axis and the axis pointing out and into the paper are maximally entangled.

this restriction |QQi can be rewritten as

+ iϕ + |QQi = cos(ϑ/2) Ψ + e sin(ϑ/2) Φ , (2.35) where

+ 1 Ψ = √ (|1i ⊗ |0i + |0i ⊗ |1i) (2.36) 2 + 1 Φ = √ (|0i ⊗ |0i + |1i ⊗ |1i) (2.37) 2 are two of the maximally entangled states (see section 2.3.2) for two qubits. An illustration can be seen in Fig. 2.3. 2.2. QUANTUM OPTICS 13

2.2.1.4 Implementation of the qubit There are different methods of implementing a qubit in practice. One convenient method for photons is to use two different orthogonal polarisations. Usually one denotes the eigenstates of σˆ3 as horizontal (H) and vertical (V)-polarised light, which is one possible implementation. Also left (L) and right (R)-polarised light, which is left-circular or right-circular polarised or the basis states of plus (+) and minus (-)-light (where the basis is rotated by 45◦ compared to H and V-polarised light) can be used, where one of the polarisations correspond to |0i and the other one to |1i. These states are, however, connected by the relations 1 |Li = √ (|Hi + i |Vi) , (2.38) 2 1 |Ri = √ (|Hi − i |Vi) , (2.39) 2 1 |+i = √ (|Hi + |Vi) , (2.40) 2 1 |−i = √ (|Hi − |Vi) , (2.41) 2 which one has to remember when chosing a basis for |0i and |1i. An illustration is given in Fig. 2.2. Another possibility is to encode a qubit into Fock-states (see section 2.2.2), so that the vacuum-state corresponds to |0i and the single-photon state to |1i, or that one uses different spatial modes where a photon in one of the two modes corresponds to |0i and a photon in the other mode to |1i. When using other particles, encodings in time, charge or spin may be useful. Particles offering several independent orthogonal properties can be used to implement more than one qubit on one particle.

2.2.2 Fock state In quantum optics it is often useful to make the number of photons in a mode the decisive property of a state. Mathematically, one uses the so called number- state representation |ni, where n is the number of photons. The vacuum state is then denoted as |0i, i.e., the state where no photons are present. Please note that I use a different notation here compared to the zero qubit-state |0i to make the destinction clear between these two states. In practice, however, one uses often the implementation |0i = |0i and |1i = |1i. If not otherwise stated this implementation will be assumed throughout the whole thesis. To make the number-state representation useful one can define the annihilation and creation operators aˆ and aˆ† as √ aˆ |ni = n |n − 1i , (2.42) √ aˆ† |ni = n + 1 |n + 1i , (2.43) 14 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS where aˆ |0i = 0 is the null vector. From the definition one can see that they create or annihilate one photon at a given state. The operators obey the commutation relation a,ˆ aˆ† =a âˆ† − aˆ†aˆ = 1. (2.44) It is also convenient to define the number operator nˆ =a ˆ†aˆ which has the property nˆ |ni = n |ni. With these relations every number state can be derived from the vacuum via 1 n |ni = √ aˆ† |0i . (2.45) n! Staying at single mode the number states are called Fock states. To collect photons with different properties into one state I will also use notations like |2H, 1Vi or |3U, 0Li, meaning 2 H-polarised photons and 1 V-polarised photon in the same mode or three photons in the upper arm of an experiment and no photons in the lower part of an experiment. When the meaning is clear from the context the subscripts may also be omitted.

2.2.3 N00N-states

N00N-states |ΨiN00N (pronounced noon-states) are quantum mechanical states consisting of an superposition of N photons in one mode and no photons in another mode and vice versa, i.e., 1 |Ψi = √ |Ni ⊗ |0i + eiϕ |0i ⊗ |Ni , (2.46) N00N 2 1 2 1 2 where ϕ ∈ R is a relative phase and the two modes 1 and 2 could be anything from different paths in an interferometer to different polarisations. Depending on what the two modes are, there are different strategies in how to produce N00N-states. Sending one photon on a 50:50-beam splitter (BS) would, for example, produce a N = 1 N00N-state where the two modes are spatial ones. N00N-states with N ≥ 2 can be used in a variety of quantum mechanical setups to improve some properties of these setups compared to their classical counterpart. Some of them will be discussed in chapter 3 and chapter 4. There are, however, problems to produce N00N-states with high N. So far, no one has succeeded in producing N00N- states deterministically at high rates for N ≥ 3, but there have been experiments reported producing N00N-states at low rates. Mitchell et al. [MLS04] or Kim et al. [KPC09] for three photon N00N-states, Walther et al. [WPA+04] for four photon N00N-states and Afek et al. [AAS10] for five photon N00N-states are some examples.

2.2.4 Coherent states Coherent states were ﬁrst described by Erwin Schrödinger in relation with the quantum mechanical harmonical oscillator [Sch26a]. A coherent state |αi is represented 2.3. ENTANGLEMENT 15 by a complex number α and is the right eigenstate of the annihilation operator [Eq. (2.42)]:

aˆ |αi = α |αi . (2.47)

If written in the photon-number representation the coherent state reads as:

2 ! ∞ − |α| X αn |αi = exp |ni . (2.48) 2 p n=0 (n!) The importance of coherent states arises from the fact that they can represent an oscillating field and hence can be used to describe the field of a continuous wave (CW)-laser (see section 2.4.1.1). The full theory of coherent states was formulated by Roy Glauber [Gla63a, Gla63b] and formed a basis for quantum optics. An often used property of coherent states is the mean photon number n¯ which is defined as

n¯ := hα| nˆ |αi = α∗α = |α|2 . (2.49)

Other important properties are the variance

(∆ˆn)2 = hα| nˆ2 |αi − n¯2 =n ¯ (2.50) and the probability P of ﬁnding n photons in the coherent state n¯n P(n) = |hn|αi|2 = e−n¯ . (2.51) n! This shows that coherent states follow Poissonian statistics. The relations Eq. (2.49), (2.50) and (2.51) show that it in many cases suﬃces to know the absolute value of α (or the mean photon number n¯) and that a knowledge of the phase of the coherent states often is not needed. This is an advantage since the mean photon number can be determined easily in experiments by intensity measurements.

2.3 Entanglement

Every separable bipartite state, that is, a state consisting of two subsystems A and B, can be written as

X + X ρˆ = piρˆAi ⊗ ρˆBi , pi ∈ R , pi = 1. (2.52) i i For pure states the deﬁnition of separability can be rewritten as

|ψi = |ψAi ⊗ |ψBi , (2.53) 16 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

where |ψAi and |ψBi, as well as ρÂ and ρˆB, are states of the corresponding system. Any non-separable state is called entangled. Similar definitions can be used for systems consisting of more than two subsystems. One talks then about multipartite entanglement, but since I will only work with bipartite entanglement in this work the above definition should be sufficient. More details could be found in my master thesis [Kot05]. Here, I will first talk shortly about how to quantify entanglement and then discuss one of the most known entangled states, the so-called Bell states. Those interested into knowing more about entanglement can, for example, read a review article written by the Horodecki family [HHHH09].

2.3.1 Concurrence The concept of concurrence C as a quantitative measure of entanglement was introduced in the papers by Hill and Wooters [HW97] and Wooters [Woo98]. First, it was derived for two qubits, but later on several (different) approaches have been done to extend the definition for higher dimensional states. For an arbitrary two qubit state ρˆ the concurrence is defined as

C (ˆρ) = max {0, λ1 − λ2 − λ3 − λ4} , (2.54) p√ ˜√ where {λi} are the eigenvalues in decreasing order of the matrix R = ρˆρˆ ρˆ and ρˆ˜ is deﬁned as ˜ ∗ ρˆ = (ˆσ2 ⊗ σˆ2)ρ ˆ (ˆσ2 ⊗ σˆ2) , (2.55)

∗ where ρˆ denotes complex conjugation and σˆ2 is the second Pauli matrix as defined in Eq. (2.32). This definition assures 0 ≤ C ≤ 1 with C = 0 for separable states and C = 1 for maximally entangled states. For pure states the definition simplifies to

C (|ψi) = ψ|ψ˜ , (2.56) where ˜ ∗ ψ =σ ˆ2 ⊗ σˆ2 |ψ i . (2.57) This deﬁnition results in the explicit formula

C (|ψi) = 2 |a00a11 − a01a10| (2.58) for pure states when one uses the notation of Eq. (2.34). A generalisation of this concurrence to any bipartite state, consisting of two n-level systems, was done by Albeverio and Fei in [AF01]. They deﬁned it as v u n−1 u n X Cn = t |aikajl − ailajk|, (2.59) 2 (n − 1) i,j,k,l=0 2.3. ENTANGLEMENT 17

n where the state of Eq. (2.34) was generalised. The fraction 2(n−1) serves as a normalisation factor, so that the concurrence remains in the range 0 ≤ C ≤ 1. In the same paper [AF01], a generalisation for multipartite states, which might consist of higher dimensional subsystems than qubits, was given. Another generalisation was given by Uhlmann [Woo00], and other generalisations are discussed in chapter 5. The concurrence is by its definition invariant under relabeling of particles and invariant under local unitary transformations. It vanishes for separable states and is in some sense easy to calculate for pure states since it doesn’t need any extremi- sations. However, for mixed states there exists no analytical formula so far (except for the special case of two qubits) and it is questionable if the concurrence is a good measure for multipartite states, since it just calculates one number whereas entanglement for multipartite states becomes a quite complex matter where most certainly higher dimensional quantities like tensors should be used to quantify entanglement. A lot of research is going on in that area and in spite of that, no widely accepted general definition of entanglement for higher dimensional systems which can relate it to some operational meaning has been found yet. For states consisting of two qubits, however, some operational meaningful measures of entanglement exist. One is the so called entanglement of formation EF which is defined for a state ρˆ as “the asymptotic number of standard singlets re- quired to locally prepare a system in [that] state” [BDSW96], where singlets will be defined soon. For two qubits, a bijective map between C and EF exists, making the concurrence a useful quantitative measure of entanglement.

2.3.2 Bell states For two qubits the four so called Bell states are deﬁned as

+ 1 Φ = √ (|0i ⊗ |0i + |1i ⊗ |1i) , (2.60) 2 − 1 Φ = √ (|0i ⊗ |0i − |1i ⊗ |1i) , (2.61) 2 + 1 Ψ = √ (|0i ⊗ |1i + |1i ⊗ |0i) , (2.62) 2 − 1 Ψ = √ (|0i ⊗ |1i − |1i ⊗ |0i) , (2.63) 2 where |Ψ−i is the so called singlet state. All these Bell states are maximally entangled and have therefore C = 1. They form an orthonormal basis for the space consisting of two qubits. Bell states have the property that a measurement of one qubit always determines the value of the other qubit. These states are widely used in quantum mechanical protocols like quantum teleportation, quantum cryptogra- phy, super-dense coding and many others (see [NC00]). Bell states may also be deﬁned for more than two qubits, but this is not part of the thesis. 18 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

2.4 Experimental components

2.4.1 Lasers Lasers are the main sources of light used in quantum optics. Some of their main advantages are that they may have a very long coherent length or small linewidth, emit highly coherent light and can have very high output power. There are many possibilities to classify lasers, but for the purpose of this work it is enough to diﬀer between CW-lasers and pulsed lasers. Information about the working principle of lasers can, for example, be found in [Sie86].

2.4.1.1 CW-lasers Lasers which continuously emit light are called continuous wave (CW). They usually oﬀer very small bandwiths of the emitted light and are especially suitable for interference experiments since they have long coherent lengths.

2.4.1.2 Pulsed lasers The opposite to CW-lasers are pulsed lasers which only emit short pulses of light. The shortest possible pulse durations produced by lasers so far are in the region of around 100 attoseconds (1 as = 10−18 s), but such short pulse durations require a lot of effort and for commercially available lasers the state of the art is in the region of 10 − 100 femtoseconds (1 fs = 1−15 s). Pulsed lasers offer high output effects which are often needed to achieve non-linear effects in quantum optics. An often used type of these lasers are titanium-sapphire (Ti:S)-lasers which can offer pulse- lengths of around 100 fs with repetition rates of around 80 MHz and an avarage power output in the order of a few Watts. Since the lasers thus only lase in the order of 1/105 of the time the peak output power can exceed 1 MW. Since the beam usually is constrained to areas of less than 1 cm2 any non-pulsed operation at these output powers would destroy materials exposed by the laser beam. Compared to CW-lasers the bandwith of pulsed lasers is much broader. The bandwith is coupled to the pulse duration and the shorter the latter the larger the former. The reason is that time and frequency are fourier coupled and one needs more and more components (frequencies) in fourier space to produce shorter pulses.

2.4.2 Nonlinear crystals Since light is an electromagnetic radiation its behaviour can be described by the Maxwell’s equations. From them one can define a quantity P~ called electrical polarisation which connects the electric field E~ to the electric displacement D~ via D~ = ε0E~ + P~ E~ , (2.64) where ε0 ≈ 8.854 As/Vm is the permittivity. If the electric field is not too high one can usually approximate P~ ≈ ε0χE~ , where χ is called the electric susceptibility. 2.4. EXPERIMENTAL COMPONENTS 19

When, however, this linear behaviour is no longer valid one talks about nonlinear optics. In that case, one usually writes

(1) (2) 2 (3) 3 P~ = ε0 χ E~ + χ E~ + χ E~ + ... . (2.65)

This relation offers a huge variety of new optical effects compared to linear optics. Depending on which of the χ(n) one has to include one talks about second-order effects if one can truncate the above expression after χ(2), third-order effects if one can truncate it after χ(3) and so on. χ(n) are, in general, tensors. In my work I will only have to deal with second order effects. Except high intensities of the light one has to have materials offering high enough values of χ(2) to be able to see nonlinear effects. This is only possible with materials not having inversion symmetry. The most important materials are so called nonlinear crystals. For an overview about nonlinear crystals and their properties and experiments done with them one can read the books [DGN99] and [Nik05]. A lot of nonlinear effects are described in books about photonics, for example [Men07]. Here, I will only focus on one effect, namely spontaneous parametric down-conversion (SPDC). Before doing that, I will talk about phase matching.

2.4.2.1 Phase matching

Second-order effects offer the possibility of converting one or two photons into one or two other photons in the crystal. For example, one incoming photon could be converted into two outcoming photons or vice versa. When there is one incoming photon it is usually denoted as pump-photon and if there are two output photons they are usually denoted as signal- and idler-photons. In all the nonlinear effects the conservation laws of physics still have to be fulfilled. By demanding conservation of energy and momenta one gets constraints about which properties (for example, polarisation, emission angle, wavelength) the output photon(s) could have for given values of the input photon(s). Details are given in the aforementioned references. Depending on the polarisation beams can be divided into ordinary (o) and extraordinary (e) beams in nonlinear crystals. These beams have, in general, different indices of refraction in the crystal and can thus propagate with different phase velocities. One divides nonlinear effects into different types, depending on how ordinary beams and extraordinary beams are arranged between input and output beams. The most important setups are type-I and type-II conversions. In type-I conversions all the input photons have the same polarisation (e or o) and all the output photons have the other polarisation (e if the inputs had o and vice versa). In type-II there are either two input photons of different polarisations (e and o) or two output photons of different polarisations. In type-I setups one can thus filter out non-converted pump-photons from converted photons by putting a polarising beam splitter (PBS) after the crystal. 20 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

(a) (b) Figure 2.4: Schematic sketch of SPDC (a), where a pump photon of wavelength λp is converted inside a nonlinear crystal into a signal and an idler photon. The wavelengths λs and λi of signal and idler, respectively, together with the direction of these photons have to fulﬁl energy and momentum conservation. In (b) a beam splitter with input modes A and B and output modes C and D is sketched.

2.4.2.2 Spontaneous parametric down-conversion In spontaneous parametric down-conversion (SPDC) [see Fig. 2.4 (a)] one pump photon is converted into one signal and one idler photon (the so-called down- converted photons), where the energies and the momenta of the two down-converted photons add to the energy and the momentum of the pump photon, i.e., neither energy nor momentum is transferred to the crystal. If signal and idler have the same frequency one talks about degenerate SPDC. Type-II SPDC emits the photons in two diﬀerent cones and if one orients the crystal such that the cones intersect at two points one can use the photons at the two intersection points to get, for example, Bell states (see section 2.3.2). Type-I SPDC emits the converted photons in only one cone where two photons originating from the same conversion process are always found at opposite sites of the cone. At the right angle of the crystal the cone may collapse into a single beam. One talks then about collinear SPDC.

2.4.3 Optical components 2.4.3.1 Beam splitter A beam splitter (BS) [see Fig. 2.4 (b) for a schematical sketch] is an optical device with usually two inputs and two outputs which redistributes the light from the input modes (A and B in the ﬁgure) to the output modes (C and D in the ﬁgure). Two of the most common practical realisations of BSs are prisms glued together or metal-covered glasses. In quantum mechanics where one adresses annihilation and creation operators [see Eq. (2.42)] to the modes (aˆ and aˆ† to mode A, ˆb and ˆb† to mode B and so on), the most general description for the operation of the beam 2.4. EXPERIMENTAL COMPONENTS 21

T T splitter is a,ˆ ˆb = B† c,ˆ dˆ , where

θ i(ψ+φ)/2 θ i(ψ−φ)/2 cos 2 e sin 2 e B = θ i(−ψ+φ)/2 θ i(−ψ−φ)/2 (2.66) − sin 2 e cos 2 e with θ, ψ, φ ∈ R. T = cos2(θ/2) and R = sin2(θ/2) are called the transmission and reflectivity, since these quantities determine which ratio of the incoming intensity is transmitted or reflected at the BS. A common convention is to choose ψ = −θ = π/2. For these values a 50:50-BS, which reflects half of the light and transmits the other half, has the simple transformation matrix 1 1 i B50:50 = √ . (2.67) 2 i 1

2.4.3.2 Polarising beam splitters A polarising beam splitter (PBS) is an optical device similar to an ordinary BS (see section 2.4.3.1), only that its transformation matrix is not given by Eq. (2.66), but depends only on the polarisation. A PBS allows one specific polarisation to pass unaffected through the PBS, whereas the orthogonal polarisation is completely reflected. As an example, a PBS which transmits H-polarised light (and therefore reflects V-polarised light) could be used to create +-polarised light at one output when fed with H-polarised light in one input mode and V-polarised light in the other input mode. (The second output will in this case only contain the vacuum mode.) A very common application of PBSs is to split light into two orthogonal polarisations.

2.4.3.3 Wave plates Wave plates are special cases of the retardation plates whose Jones matrix is given by Eq. (2.9). As can be seen from the Jones matrix they change the polarisation of the light travelling through them. The two most important kind of wave plates are half-wave plates (HWPs) and quarter-wave plates (QWPs). A HWP has φ = π in Eq. (2.9), leading to the general formula (when ignoring an overal phase-factor) cos(2θ) sin(2θ) M θ = (2.68) HWP sin(2θ) − cos(2θ) when it is oriented with an angle θ to the x-axis. When using linear polarised light the eﬀect of a HWP is to change its polarisation angle by 2θ. When using circular polarised light a HWP turns left circular polarised light into right circular polarised light and vice versa, irrespective of θ. A QWP has φ = π/2 and thus its general Jones Matrix (ignoring again an overall phase-factor) is i + cos(2θ) 2 cos(θ) sin(θ) M θ = (2.69) QWP 2 cos(θ) sin(θ) −i + cos(2θ) 22 CHAPTER 2. BACKGROUND IN OPTICS AND QUANTUM OPTICS

It converts linearly polarised light into elliptically polarised light and vice versa. A special case occurs when θ = π/4. In that case, linearly polarised light is converted into circularly polarised light (H into R, V into L) and vice versa.

2.4.3.4 Polarisers A polariser is an optical component which transmits only light with a speciﬁc polarisation. For a linear polariser its Jones Matrix is given by Eq. (2.12) in connexion with Eq. (2.13). Another type of polarisers are circular polarisers whose Jones matrices are given by

1 1 i M r = (2.70) pol 2 −i 1 for a right circular polariser and its adjoint for a left circular polariser. One of the main applications of polarisers is to assure that one has light with a given polarisation. By rotating the polariser around the optical axis and measuring the intensity of the transmitted light one can on the other hand take conclusions about the polarisation of the light before the polariser. The main diﬀerences between polarisers and wave plates (see section 2.4.3.3) are that the latter transmit all the light, while the former absorb light of the “wrong” polarisation and that the polarisation of the light passing a polariser is independent of the incoming light, while the polarisation of the light after a wave plate depends on the polarisation of the incoming light.

2.4.3.5 Dichroic mirrors Dichroic mirrors are mirrors which reflect (partially or fully) light of a certain wavelength (or in a certain range of wavelengths) and transmit light of other wavelengths. Dichroic mirrors can thus be used to separate light of different wavelengths. Since they can be made non-absorbing they can be used for light of high intensity which would destroy absorbing mirrors. The fraction of the reflected and transmitted light depends usually on the incidence angle.

2.4.4 Detectors There exist a lot of different strategies for detecting photons. The detectors differ widely in which intensity range they can detect, how much dead time they have after a detection, which wavelengths they can detect, how efficient they can detect photons and so on. For experiments in quantum optics one often uses single-photon detectors (SPDs), i.e., detectors, which can detect as low intensities as one single photon per unit counting time. One can, however, usually not determine whether one has detected one, two, or more photons per unit counting time, although much effort is invested by some research groups to achieve just that. 2.4. EXPERIMENTAL COMPONENTS 23

A special kind of detector is avalanche photodiodes (APDs). They are semi- conductor devices which work similarly to a photomultiplier, i.e., one impinged photon initiates a shower of other electron-hole pairs which then can be detected. A detailed overview of such detectors can be found in [Cam07]. An important quantity when detecting photons in, for example, interferometers is the visibility. If Imax denotes the maximum intensity and Imin the minimum intensity in some experiment, then the visibility V can be deﬁned as I − I V = max min . (2.71) Imax + Imin

For intereferometric fringes Imax and Imin could, for example, denote the intensity at a peak and in a valley of the fringe, respectively. Often in experiments one has the case that the intensity is proportional to the number of detected photons N and one could then replace I by N in the above formula.

Chapter 3

Phase resolution and sensitivity

In this chapter I will discuss phase super-resolution and phase super-sensitivity. I will introduce these concepts in separate sections. For phase super-resolution, I will show some of the experiments done so far for achieving it. First, I will show an experiment requiring entanglement (which, until some years ago, was believed to be a necessary ingredient for phase super-resolution), then I will show an experiment which does not need entanglement, but introduces a concept called time-reversal symmetry to achieve phase super-resolution. Finally, I will report on an experiment done by myself to show that neither entanglement nor joint detection (nor anything such as time-reversal symmetry) is needed to achieve phase super-resolution, and that by dropping theses constraints one is able to achieve an arbitrarily high phase super-resolution and much better visibilities than obtained in previous experiments. I will only discuss phase super-resolution with photons. However, experiment using other particles like ions exist [LKS+05]. It is not possible to achieve phase super- sensititvity with the experiment I performed. This will be shown in section 3.2.

3.1 Phase super-resolution

For explaining phase super-resolution one can start by looking at a simple example, such as a Mach-Zehnder-interferometer [Mac92, Zeh91] as depicted in Fig. 3.1. Photons enter the interferometer through the ports A and B. The two modes interact at a 50:50-beam splitter (BS) and proceed as mode C and D through two arms of equal length to another 50:50-BS, where they interact again and leave in modes E and F. In one of the arms a phaseshifter (PS) is placed so that a phaseshift acts on mode C. Finally, single-photon detectors (SPDs) are placed in mode E and F to detect the photons arriving there. Imagine that one photon at a time (i.e., the state |1i) arrives in mode A, and zero photons (i.e., the vacuum state |0i) is present in mode B. Due to the ﬁrst BS

25 26 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY

Figure 3.1: A setup with a Mach-Zehnder interferometer where a phaseshifter (PS) is inserted into one of the arms. this transforms [see Eq. (2.67)] to

1 1 |ΨiMZ = √ (|1iC |0iD + |0iC |1iD) (3.1) BS1 2 after the BS and to 1 |Ψi1 = √ eiϕ |1i |0i + |0i |1i (3.2) MZPS 2 C D C D after the PS. The ﬁnal BS then gives

1 1 iϕ iϕ |ΨiMZ = 1 + e |1iE |0iF + 1 − e |0iE |1iF (3.3) BS2 2 and monitoring the output of mode E with the SPD (see section 2.4.4) gives 2 hΨ|1 |1i |0i = cos2 (ϕ/2) (and sin2 (ϕ/2) for mode F) as the probability MZBS2 E F to detect a photon [see Fig. 3.2 (a)]. In other words, the probability of detecting one photon undergoes one oscillation when changing the angle ϕ of the PS from 0 to 2π. If one changes the setup, so that now one photon is present in both mode A and B, and that they are prepared so that they are indistiguishable, after passing 3.1. PHASE SUPER-RESOLUTION 27

1 mode E mode F 0.8

0.6

0.4

0.2

probability for 1-photon detection 0 0 π/2 π 3π/2 2π ϕ (a) 1 2 in mode E (F) 0.9 1 in each mode 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 probability for 2-photon detection 0 0 π/2 π 3π/2 2π ϕ (b) Figure 3.2: The upper picture (a) shows the probability for detection of one photon in mode E (F) with |Ψi1 as the input state, the lower picture (b) shows the MZBS1 probability of coincidence detection between one photon in mode E and one photon in mode F (and the probability of detecting two photons in mode E or two in mode F) with |Ψi2 as the input state. MZBS1 28 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY the ﬁrst BS one will get

2 1 |ΨiMZ = √ (|2iC |0iD + |0iC |2iD) , (3.4) BS1 2 i.e., the N00N-state with two photons [see Eq. (2.46)]. After the PS one gets the state 1 |Ψi2 = √ ei2ϕ |2i |0i + |0i |2i (3.5) MZPS 2 C D C D and, finally, √ 2 1 h i2ϕ i2ϕ i |ΨiMZ = √ 1 + e (|2iE |0iF + |0iE |2iF) + 2 1 − e |1iE |1iF (3.6) BS2 2 2 after the second BS. If one now monitors the cooincidences between mode E and 2 F with the SPDs (i.e., hΨ|2 |1i |1i ) one will get sin2(ϕ) as the result for MZBS2 E F the probability of detecting one photon in each mode [see Fig. 3.2 (b)]. When changing the angle ϕ of the PS from 0 to 2π this gives now two oscillations, i.e., twice as many as in the case when only one photon was present in the interferometer. This property of showing 2 oscillations over a period which normally gives only one oscillation is called 2-fold phase super-resolution. The same effect would have happened when one would just have looked at two-photon events in mode E (F) when the state |Ψi2 is present, only that one would have needed photon MZBS2 number-resolving detectors (PNRDs) instead of SPDs in my setup. When a “high N00N”-state (i.e., N > 2) is present in mode C and D and one would monitor the probability of finding all n = N photons in the same output mode after the second BS as a function of ϕ, one would be able to see n oscillations over the same period of the angle ϕ and therefore n-fold phase super-resolution.

3.1.1 Experiments with entanglement Two of the most important experiments for showing phase super-resolution are those done by Mitchell et al. [MLS04] for three photon N00N-states and by Walther et al. [WPA+04] for four photon N00N-states . (Another important experiment done by Nagata et al. [NOO+07] is discussed in more detail in section 3.2.) In the Mitchell et al. experiment, two photons from a spontaneous parametric down-conversion (SPDC)-process with a type-II β-barium borate (BBO) crystal which is pumped by light from a femtosecond (fs) titanium-sapphire (Ti:S) laser are combined into the same spatial mode by a polarising beam splitter (PBS). A half-wave plate (HWP) and partial polariser (PP) shift the polarisation to ±60◦ and then a local oscillator (LO) photon (taken from the pump beam) is added to the same spatial mode. Finally, a quarter-wave plate (QWP) is used to produce the three-photon N00N-state in the horizontal (H)-vertical (V) basis (see Fig. 1 in 3.1. PHASE SUPER-RESOLUTION 29

[MLS04] for a more detailed description). At the end the states are analysed in the

|2, 1i±45◦ and the |3, 0i±45◦ basis, where the subscripts denote the polarisation. In both cases, phase super-resolution was achieved. However, the count rates were very low (on the average around 1.3 photon coincidences per second after background subtraction in the first case, and around 0.17 photon coincidences per second after background reduction in the latter case). Their achieved visibility (for the detection in the |2, 1i±45◦ -base) was 42 ± 3%. In the setup of Walther et al., two Mach-Zehnder interferometers (see Fig. 3.1) are used. The beam from a fs Ti:S laser passes a type-II BBO crystal which, by SPDC, produces photon pairs, where one of the photons is sent to an arm in one interferometer and the other photon into an arm in the other interferometer. The pump beam is then backreflected by a mirror onto the BBO-crystal and produces two more photons, which are sent into the other arms in the respective interferometers (see Fig. 1 in [WPA+04] for a more detailed explanation). Before the SPDs, polarisers oriented at 45◦ are placed, and the interference fringes are observed by moving the backreflection mirror of the pump. In their experiment, Walther et al. could on average achieve approximately 0.05 photons per second in coincidence and a visibility of around 61%.

3.1.2 Experiments without entanglement In 2007 it was shown by Resch et al. [RPP+07] that entanglement is not needed for phase super-resolution. According to their paper, the only thing which was needed was time-reversal symmetry which is a special implementation of a more general measurement technique introduced by Pregnell et al. [PP04, Pre04]. The concept of time-reversal symmetry is based on the fact that probabilities are invariant under time reversion. Therefore, instead of creating N00N-states, one needs only to detect N00N-states, which can be done by coincidence counting with SPDs. The state creation in that scheme is done by using two orthogonal polarisation modes of a laser (H and V in their paper) where they assure equal amplitudes for them and by having a vacuum mode as one of the inputs in their BSs. More details can be found in [RPP+07], but it is not important to fully understand the concept of time-reversal symmetry, since this is not needed for showing phase super-resolution, as will be clear from the next section. The important features from Resch et al. experiment are counting rates of an average of around 18 photons per second in coincidence for three-fold phase super-resolution (5 photons per second for four-fold and 1.3 photons per second for six-fold resolution) and visibilities of 91 ± 3, 76 ± 2 and 90 ± 2%, respectively.

3.1.3 Experiments without entanglement or joint detection I shall now show that neither entanglement nor joint detection (or time-reversal symmetry) is needed for achieving phase super-resolution. To realise that, one can 30 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY

1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0 π/2 π 3π/2 2π ϕ sin(ϕ) sin(ϕ+3π/4) 3 3 sin(ϕ+π/4) 2 ⋅∏m=0sin(ϕ+mπ/4) sin(ϕ+π/2)

Figure 3.3: Illustration of Eq. (3.7) for n = 4. One can see that the product of the phase-shifted sine functions performs 4 times as many oscillations as the functions themselves. look at the simple relation sin(2ϕ) = 2 sin(ϕ) cos(ϕ), or, more generally,

sin (nϕ) π 2π (n − 1)π = sin (ϕ) sin ϕ + sin ϕ + ··· sin ϕ + .(3.7) 2n−1 n n n This relation shows that a sine-function with n oscillations over a period of ϕ, which ordinarily would give only oscillation, can be replaced by a product of n sine-functions with ordinary oscillation-periods, but each of them phase-shifted by a factor of π/n. In Fig. 3.3 an example for n = 4 is given. To use this relation in experiments it is very convenient to use coherent states (see section 2.2.4), since they have the property of beeing still coherent states after having passed a BS (see section 2.4.3.1 and Fig. 3.4). It also makes no diﬀerence whether the two half beams come from the same source, or from two diﬀerent, but identical sources. This opens the possibility to choose, quite freely, how the states are splitted. One does not need to split the state after having passed the object (simulated by the PS in Fig. 3.1) like in [RPP+07] to keep time-reversal symmetry. Instead one can choose to split the state before the object or in time or in frequency instead. In any case the splitted states will keep the property of still beeing coherent. One only has to assure that the object under investigation acts on the beam in the same way under the whole irradiation, i.e., it is not changing its behaviour over the width of the beam, the time duration of the experiment, or the range of frequencies used. Dropping the requirement of time-reversal symmetry, one 3.1. PHASE SUPER-RESOLUTION 31

Figure 3.4: A coherent state split spacially into two by a mirror after the object.

also gets rid of joint measurements, thereby reducing the number of components needed. I did two experiments to implement these ideas, one in the space-domain and one in the time-domain, thereby showing that neither entanglement nor time-reversal symmetry (in the latter case also no joint-detection) is needed to achieve phase super-resolution. Doing them, I could, to the best of my knowledge, achieve much higher count-rates and much better visibility than in any of the experiments done so far.

3.1.3.1 Experiment in the space-domain A sketch of the experiment in space-domain can be seen in Fig. 3.5. I used a pulsed laser with a wavelength of λ = 1550 nm to create the coherent state. The repetition rate was 2.5 MHz and the pulse duration around 500 ps. The laser had an average power of 1 mW, which, immediately after the laser, was attenuated by 50 − 55 dB (not shown in the figure) to not destroy the sensitive SPDs. The PBS after the laser assures that the beam is H-polarised. Then the beam is splitted by two 50:50-BSs, which are oriented in such a way that the beam falls on the object (simulated by the first HWP). A mirror reflects the remaining beam onto the object as well. The angle α = 2.9◦ between one of the outer and the middle beam is small enough to assure that the object affects the three beams in essentially the same way. In the case where the object is a HWP this means that the optical path length through the HWP is basically the same for all three beams. The HWP can be controlled by a computer which can orient it at any angle ϕ to give a phase-change between the H- and V-polarised modes. Since a turn of the HWP by ϕ = 45◦ transforms the polarisation from H to V [see Eq. (2.68)], a turn of 90◦ would bring back the beam to its original polarisation. When monitoring the intensity with a detector one would have then seen one whole oscillation (corresponding to 360◦, or 2π, in all my figures). A PBS in the middle beam assures that the beam is splitted into an H- and a V-polarised part, which are subsequently detected by SPDs then. Two HWP, 32 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY

Figure 3.5: Setup to measure six-fold phase super-resolution in space-domain. See the text for further details.

one oriented at an angle of 15◦ in one of the other beams and one oriented at 30◦ in the other beam, together with a PBS and SPDs in each beam, assure that one has six beams at the SPDs, each shifted equally according to Eq. (3.7) (remember that 1◦ in the setup corresponds to 180/π in the equation). The output of the SPDs are then coupled into single-mode ﬁbres (SMFs) and sent to a six-channel coincidence unit. The SPDs are gated simultaneously with time periods of 1 ns and the coincidence unit sends the output of all the SPDs and the coincidence events during the gating time between any of its channels to a computer. The result of the experiment can be seen in Fig. 3.6. The grey dots correspond to the six-fold coincidence events detected by the SPDs and the coincidence unit. The scale to the left applies and the statistical errors (calculated as the square roots of the counting rate) are smaller than the dot size. One can clearly see that a change of the HWP of 90◦ (corresponding to 360◦ in the ﬁgure) shows sixfold oscillation over the period which usually would have shown only one oscillation, thereby demonstrating phase super-resolution. At every angle, counts were collected for 10 seconds giving an average counting rate of around 750 photons per second. The visibility (calculated by taking the lowest of the two minima around each peak) is between 97% and 98.6%, outranging the visibility of all previous experiments. At two of the minima the counting rate is a little bit higher than at the other four valleys. This is due to the fact that one PBS did not split the beam perfectly into H- and V-polarised parts. 3.1. PHASE SUPER-RESOLUTION 33 L L 20 000 per 10s H arb. units 15 000 H

10 000

5000 fold coincidence counts

- 0 0 50 100 150 200 250 300 350 Multiplied counting rate Six Phase shift HdegreesL

Figure 3.6: Demonstration of six-fold phase super-resolution in space-domain. The grey dots corresponds to coincidence measurements (left scale applies) and the black dots to the case where the single counting rates were multiplied (right scale applies).

Instead of looking for coincidences, one could also get phase super-resolution by multiplying the counting rates of all the SPD. The result can be seen as the black dots in Fig. 3.6, where the right scale applies. Here, the visibility is between 98.6% and 96.7%. Further improvements of the setup could be done by changing the non-optimal PBS or by replacing the ﬁrst 50:50-BS with a 33% reﬂecting BS. The latter would not, a priori, change the visibility, but increase the coincidence rate. The setup also allows a scaling to more than six fringes by adding more “arms” (beams) to the setup. The only limiting factor is that the angles between the outermost beams should not get too large as described above. However, a disadvantage would be that more fringes require more components, soon making the setup bulky and expensive.

3.1.3.2 Experiment in the time-domain To overcome the limitations of the space-domain setup, one can perform the measurement in the time-domain as depicted in Fig. 3.7. From the laser to the HWP it is exactly the same setup as in Fig. 3.5, only that the BSs are removed. The first HWP simulates still the object. But instead of splitting the beam spatially one can split the beam temporally. This is done by the second HWP in the setup which is also controlled by the computer. For the sake of simplicity one can think of the setup in Fig. 3.7 without the coincidence unit and with only one detector. For getting for example 10 fringes one counts first the photons when the first HWP is turned from 0◦ to 90◦ in small steps, then one turns the second HWP by 90◦/10 = 9◦ and counts 34 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY

Figure 3.7: Setup to measure phase super-resolution in time-domain. See the text for further details.

the photons again while turning the first HWP in small steps, and so the procedure continues until the second HWP has been turned by 90◦. By doing so, one has transferred the arms, or beams, from the setup in space-domain into different arms “after each other” in time-domain. The values of the measurements are stored and finally multiplied. The results can be seen in Fig. 3.8 (a). One can clearly see the 10 fringes. At every step the photons were counted for one second. To make the measurement taking only half the time one can take advantage of the second SPD and the coincidence unit, since the second output of the PBS corresponds to the situation when the second HWP is changed by 45◦ compared to the first output, and one therefore only has to turn the second HWP by 45◦ instead of 90◦. The results of the setup with two SPDs can be seen in Fig. 3.8 (b). This second method gives visibilities of up to 99.6%. By lowering the angle increment of the second HWP one can get more fringes, an example for n = 30 is shown in Fig. 3.9. One should note that the x-axis now spans a shorter range. The reason that the visibility is worse is most certainly due to the fact that the laser which was used is not stable over long times. This causes measurement problems since it is impossible for the detector to determine whether a changed counting rate is due to a changed intensity of the laser or due to a phase-change at the object. As an example for the n = 30 case I measured one second at every point, but I had 30 steps of the second HWP and 1800 steps of the first HWP at each turn (to give 60 points per fringe), which makes a total measurement time of around one day, when one includes the time for turning the HWPs. The advantage of the time-domain method lies in the fact that it can be scaled, 3.1. PHASE SUPER-RESOLUTION 35

L 500

400 arb. units H

300

200

100

Multiplied counting rate 0 0 50 100 150 200 250 300 350 Phase shift HdegreesL (a) L

200 arb. units H 150

100

Multiplied counting rate 0 0 50 100 150 200 250 300 350 Phase shift HdegreesL (b) Figure 3.8: Demonstration of ten-fold phase super-resolution in time-domain by using one SPD (a) and by using two SPDs and a coincidence unit (b). 36 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY

L 120

100 arb. units H 80

Multiplied counting rate 0 0 10 20 30 40 Phase shift HdegreesL

Figure 3.9: Demonstration of 30-fold phase super-resolution in time-domain. Notice that the x-axis only spans the range between 0◦ and 45◦.

in principle, to arbitrary numbers of n. One only has to use smaller steps of the HWP. This scaling up can furthermore be done without any extra components. In practice the measurement time will most probably not be a big issue, as the ﬁrst HWP is an object under investigation where one is only interested in monitoring changes in the phase and not in measuring all 1800 points over a full 0−360◦ range from it under each measurement increment of the second HWP. Furthermore, more stable lasers than the one I used, exist on the market. The only limitation in the setup is the step size of the HWPs-holder. In my experiment I had rotation- holders for the HWPs allowing increments of under 1 minute of arc, but even better holders for HWPs might exist. A more fundamental limitation is the demand that the object under investigation should not change its behaviour (in respect of how to inﬂuence the polarisation of the beam) during one measurement period.

3.1.3.3 Wavelength

The experiment was performed at a wavelength of 1550 nm, in contrast to previous experiments, which were usually done at a shorter wavelength. Using the telecom wavelength, together with the fact that one can use coherent beams which don’t suﬀer much from photon losses, and with the fact that one can split the beam quite freely, opens the possibility for remote applications, since the source, the object and the detectors can be placed at diﬀerent locations. 3.2. PHASE SUPER-SENSITIVITY 37

3.2 Phase super-sensitivity

With “classical” light, for example coherent states [see Eq. (2.48)], one can determine an unknown phase only up to a speciﬁc uncertainty ∆ϕ, the so called standard quantum limit (SQL):

1 ∆ϕSQL ≥ √ , (3.8) n¯ where n¯ is deﬁned as in Eq. (2.49). Using a deﬁnite number N of photons in another (unentangled) state one could replace n¯ with N and the above inequality still holds. The limit presented in Eq. (3.8) is, however, not the limit imposed by quantum mechanics. The quantum mechanical limit is

1 ∆ϕ ≥ , (3.9) H N when using N entangled photons. It is also called the Heisenberg limit. An experiment is said to have phase super-sensitivity when the phase uncertainty is smaller than allowed by the SQL of Eq. (3.8). So far, phase super-sensitivity has been reported for up to N = 4 [NOO+07]. In their experiment, Nagata et al. use a so called displaced Sagnac interferometer which is more stable than a Mach-Zehnder interferometer. However, the calculations are the same as for the Mach-Zehnder interferometer of Fig. 3.1 and therefore there’s no need to introduce the Sagnac interferometer here. Nagata et al. used

|22iAB as the input state, which means two photons in each mode. This does not produce a N00N-state after the ﬁrst BS, but rather a superposition of a N00N- state and the state |22iCD. When, however, detecting the output modes |31iEF and |13iEF they will be only produced by the terms of the N00N-state in the interferometer and not by the |22iCD-term. By detecting these terms Nagata et al. were able to achieve phase super-sensitivity. To do that they had to achieve visibilities [see Eq. (2.71)] above a certain value, but I will not discuss that here.

3.2.1 Simple approach As an example for the uncertainty limit one can look at the interferometer represented in Fig. 3.1, where one now could use a coherent state |αi with α 0 as the input in mode A and the vacuum state |0i as the input in mode B. One could analyse how the state performs through the system of BSs and PS. When finally detecting nˆdiff =n ˆF − nÊ, i.e., the difference between the photon numbers 2 in the two output modes, one gets on the average hnˆdiff i = |α| cos(ϕ). Looking for 2 2 simplicity at the case ϕ = π/2 one can show that (∆ˆndiff ) = |α| =n ¯ (this can be seen by writing nˆdiff as a function of the operators of mode A and B) and the partial derivative becomes |∂ hnˆdiff i /∂ϕ| =n ¯. From that the phase uncertainty is 38 CHAPTER 3. PHASE RESOLUTION AND SENSITIVITY easily calculated to

∆ˆndiﬀ 1 ∆ϕ = = √ . (3.10) ∂hnˆdiﬀ i n¯ ∂ϕ

For other values of ϕ the uncertainty might be higher. In general, one can deﬁne a sensitivity S as

∆ϕ 1 S(ϕ) = SQL = √ , (3.11) ∆ϕ N∆ϕ or n¯ instead of N depending on the kind of experiment. S > 1 then implies that one has phase super-sensitivity. This is a simple approach and when doing it one should include among other things both the detection eﬃciency η and the visibility V when calculating ∆ϕ. Usually ∆ϕ is a function of the detection probability P . In the case of my experiment one has

∆P (n) ∆ϕ(n) = , (3.12) ∂P (n) ∂ϕ where P (n) denotes the probability of detecting n photons in coincidence. It has often been assumed that S(ϕ) takes its maximum value at the same phase when the slope of P takes its maximum. However, Okamoto et al. [OHN+08] showed that this is, in general, not the case.

3.2.2 Fisher-information

For a more detailed study on the phase uncertainty one should involve the concept of Fisher information (see for example [Fri04] for an introduction) and quantum estimation theory. These theories can tell one how to construct operators that allow one to extract the maximum amount of information out of a system and how to maximise its accuracy. However, since it seems that entanglement is needed to achieve phase super-sensititvity [NOO+07] and since the simple approach discussed above shows that one indeed does not have phase super-sensitivity (see below), I will not discuss the concept of Fisher-information any further here. I will, however, reference to the paper of Berry et al. [BHB+09], where they discuss the question how one can perform the most accurate possible phase measurements with N00N- states and how one can avoid the ambiguity which arises since one has several fringes, similar to for example Fig. 3.6 or Fig. 3.8 in this thesis. 3.3. SUMMARY 39

3.2.3 Experimental results When using the simple approach as in section 3.2.1 one has    n/2 2 |α|2 cos2 ϕ/2 + 4(k−1)π Y n P (n) = (1 − exp −   n  k=1    n/2 2 |α|2 sin2 ϕ/2 + 4(k−1)π Y n × (1 − exp − (3.13)   n  k=1 in my experiment, where n is even. A motivation is given in the paper C which is part of the thesis. Here, both η = 1 and V = 1 is implicitly assumed as will be motivated later. In my experiment the sensitivity depends both on the number of photons detected in coincidence and on the phase, i.e., S(ϕ, n). By doing the calculations it turns out that S is maximised for the phases ϕ = 0, 2π/n, . . . , 2(n − 1)π/n. It also turns out that the higher n¯ the higher the sensititvity. Taking the limit one gets

r 2 lim S(n) = (3.14) n¯→∞ n for even n and maximised over all possible values of ϕ. I could show this explicitly for n ∈ {2, 4, 6, 8}. For higher n the expressions for the sensitivity before taking the limit get quite messy, but I would conjecture that Eq. (3.14) holds for any even n. Taking the other limit one gets

lim S(n) = 0 (3.15) n¯→0 when maximising over all possible ϕ again. This shows that S ≤ 1 for all possible values of n, n¯ and ϕ and one therefore cannot see phase super-sensitivity in that kind of experiments. Since more realistic V < 1 or η < 1 would result in even lower sensitivities one does not need to pay attention to them here.

3.3 Summary

I have shown that phase super-resolution is a purely classical eﬀect. To achieve it, neither entanglement, time-reversal symmetry, or joint detection have to be used. By dropping these constraints one has the chance to think about other ways to achieve phase super-resolution experimentally. There is, however, nothing that points out that entanglement is not needed for phase super-sensitivity. To achieve that for high values of n remains very diﬃcult.

Chapter 4

Quantum imaging and lithography

Quantum imaging and quantum lithography are technologies that use the laws of quantum mechanics to circumvent the resolution limits of classical optics. First, I will describe the theoretical idea behind it in section 4.1, then I will describe some important experiments done in the field in section 4.2. Although it has been shown that one can implement the ideas of quantum imaging and quantum lithography in proof-of-concept experiments, an unsolved question so far has been to what extent the idea is efficiently scalable to any practical applications. In section 4.3 I will first present two different theories about the scaling efficiency. Then I will present an experiment which gives different outcomes for these two theories and therefore can give a hint about the scaling behaviour of the efficiency of quantum lithography. I have tried to perform such an experiment, but another group succeeded doing a similar experiment before us. Their experiment was done for other reasons, but from their results one can derive the results I was looking for. In section 4.3.4 I will, after the presentation of the idea of the experiment, describe the realisation of it by Peeters et al. and then do a thorough mathematical analysis of the experiments mentioned. I will subsequently describe in section 4.4 what one should consider when performing a similar experiment and try to analyse some difficulties of performing it. Finally, I will conclude this chapter in section 4.5 by summarising the results and giving an outlook about what one can do in the future and which questions are still unsolved.

4.1 Theoretical background

4.1.1 The classical limit Classically, the resolution of any imaging system is limited by the so called Rayleigh limit. A detailed treatment of it is given, for example, in [BW99], so I will only give a short motivation of it here. Imagine the following situation: two monochromatic point sources emit light which goes through a diﬀraction limited imaging system

41 42 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY intensity intensity

position on the screen position on the screen

point 1 point 1 point 2 point 2 sum of point 1 and 2 sum of point 1 and 2 (a) (b) Figure 4.1: Intensity distributions of two monochromatic uncorrelated point sources when the light has travelled through a diﬀraction limited imaging system with a circular lens. The intensity and the position are in arbitrary units. In (a) the sum of the intensities does allow to resolve two points, in (b) the sum of the intensities does not. with a circular lens. Putting a screen at the end of the imaging system, each of the two point sources will produce a pattern like that in Fig. 4.1 on the screen. Since I assume the point sources to be uncorrelated, the total intensity will be the sum of the intensity distributions of the two point sources. This is indicated in the ﬁgure as well. In the left plot the distance between the two intensity distributions is large enough to give two peaks to the sum of the intensities, thus making the two point sources “visible” at the screen. If the two point sources are too close to each other as in the right plot, there will be only one peak in the sum of the intensities and one has therefore hardly no longer a possibility to guess that this pattern originates from two point sources rather than one. The Rayleigh criterion states roughly that the images of two points have to be far enough apart to still give two peaks in the resulting picture (the intensity sum) to make them resolvable. A quantitative analysis was given by Ernst Abbe [Abb73], who was not believing statements of some English manufactors at that time who claimed that they had constructed extremely powerful microscopes. Af- ter a thorough investigation he argued that the performances of their microscopes should be practically impossible and formulated the classical limit “...so folgt, dass, wie auch das Mikroskop in Bezug auf die förderliche Vergrösserung noch weiter ver- vollkommnet werden möchte, die Unterscheidungsgrenze für centrale Beleuchtung doch niemals über den Betrag der ganzen, und für äusserste schiefe Beleuchtung niemals über den der halben Wellenlänge des blauen Lichts um ein Nennenswerthes hinausgehen wird.”, i.e., at the highest possible inclination angle only details in the order of λ/2, where λ is the wavelength of the used light, can be resolved. Al- though Abbe’s investigations were done for microscopes, they are equally valid for lithography schemes. 4.1. THEORETICAL BACKGROUND 43

4.1.2 How to overcome the classical limit Since the classical resolution limit is proportional to the wavelength of the used light, one might wonder why one is not simply using light of shorter wavelengths to increase the resolution. The reason is that light of shorter wavelength has higher energy per photon and that this light, being ultraviolet (UV)-light, X-rays or γ- rays, might harm or destroy the system to investigate. Furthermore, one has to apply extra security measures to not harm persons in the surrounding. The most practical reason, is, however, that shorter wavelengths can increase the cost by several orders of magnitudes since the light sources are much harder to produce and, more seriously, one gets at some point problems to find any transparent materials at all for that wavelength and having at the same time a significantly high difference in their index of refraction compared to air or vacuum to use this material as a lens. Therefore, it is both practically and economically important to find other ways of generating or imaging small patterns. I will not address any scanning schemes or schemes putting restrictions on the samples (for example schemes that only allow extreme thin samples) like electron or atomic microscopy, since they are only usable in special cases and don’t allow any general large scale imaging or lithography schemes.

4.1.2.1 Using “classical” schemes A “classical” scheme to circumvent the Rayleigh limit was proposed by Yablonovitch and Vrijen [YV99]. They use the fact that the two-photon absorption rate contains terms which are proportional to twice the frequency of the one-photon absorption rate. By introducing zone plates on lenses in the imaging system they show that one in theory could make frequency modulations which would force the terms oscillating at the one-photon absorption rate to smear out in time to a constant background, only leaving a pattern of twice the ordinary frequency, i.e., half the classical limit. They also suggested a method how one could potentially get rid of the background if some new materials with speciﬁc properties are developed. Another system with limited visibility is presented by Bentley and Boyd [BB04] and some kind of overview of the developments in this area are given in [BB06] and the references therein, but all these schemes can’t simultaneously produce or image a sub-Rayleigh feature size and sub-Rayleigh feature seperation, at least not until one has developed some new kinds of detectors [OKS94, HMSZ06, SHZ07].

4.1.2.2 Using “quantum” schemes A “quantum” scheme which allows one to surpass the classical limit, and simultaneously allowing perfect visibility, was proposed by Boto et al. [BKA+00]. In their proposal they used an interferometer similar to the Mach-Zehnder interferometer in Fig. 3.1, only that a screen is placed at the position where the second beam splitter (BS) is placed in the ﬁgure. Therefore, one has the state of Eq. (3.5), or, 44 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY in general, when having a N00N-state in the interferometer, 1 |ΨiN = √ eiNϕ |Ni |0i + |0i |Ni (4.1) screen 2 C D C D at the screen. When having an N-photon absorbing material at the screen and using gracing incidence, instead of an incidence of 45◦ as in Fig. 3.1, they calculate the deposition function ∆N as

∆N = 1 + cos(2Nϕ), (4.2)

cl compared to ∆1 ∝ 1 + cos(2ϕ) for “classical” light when using 1-photon absorption or to

cl clN ∆N ∝ ∆1 = c1 + c2 cos(2ϕ) + O(cos(4ϕ)) (4.3) with some constants c1, c2 ∈ R, when using N-photon absorption. Since one can relate the phase shift to the oﬀ-centre position x at the screen via ϕ = x2π/λ, one can see that ∆N oﬀers a resolution of λ/(2N), compared to the classical Rayleigh limit of λ/N. One should note that this scheme really requires N-photon absorbing materials. Allowing even absorption of M < N photons will limit the resolution to λ/(2M) instead.

4.2 Experiments

Since the original proposal of Boto et al. some experiments have been performed which show a better resolution than given by the classical limit.

4.2.1 Experiments with interferometers An experimental realisation which is close to the original proposal of Boto et al. was done by Chang et al. [CSOHB06]. They use an interferometric set-up with acrylic glass as the recording medium. For N = 2 they achieve a fringe separation of 213 nm when using beams of λ = 800 nm, a seperation roughly half as long as allowed by the Rayleigh limit.

4.2.2 Experiments with slits and gratings The ﬁrst experiment demonstrating the proposal of Boto et al. using slits was done by D’Angelo et al. [DCS01], a similar setup was also used by Shih [Shi03]. Later on, Shimizu et al. performed a similar kind of experiment with a grating [SEI03, SEI06]. When comparing their setups with the concluding experiment presented in section 4.3.4, one sees that all the groups, by making small changes of their setup, would have been able to answer the question about the scaling 4.3. EFFICIENCY 45 behaviour which I address in section 4.3.3. Since they, however, did not address the question, I can name another experiment using double-slits, namely the experiment of Peeters et al. [PRv09] which was done in another context, but that has important implications to quantum imaging, as will be shown later.

4.2.3 Further experiments Some of the experiments using other setups are those of Giovanetti et al. [GLMS09] and Pe’er et al. [PDV+04]. I will, however, not go into any details with them here but give them as further references.

4.2.4 Known limits Although quantum imaging and quantum lithography allows one to overcome the classical resolution limit (see section 4.1) and allows, in principle, arbitrarily small resolutions, there are some technical challenges which have to be solved. One is that one has to find a material which changes its behaviour only when absorbing N photons at the same time [YV99] and not when absorbing k photons (k < N) or when absorbing first k photons, and subsequently N − k photons. In [CSOHB06] Chang et al. demonstrate an experimental realisation of quantum lithography for N = 2 when using acrylic glass as the absorber. However, a material for more general schemes has still to be found. Agarwal et al. [ACB+07] did theoretical calculations addressing the desireable features of such materials, and Plick et al. [PWAD09] showed that the absorption rates of those materials have to be optimised. It was found out by Tsang [Tsa08], however, that there are, unfortunately, limits to which extents this is feasible. Another open question is how to produce the states needed for quantum lithography in the optimal way [GCD+08, SVD+08], or, for the case of N00N-states and N > 2, how to produce them efficiently at all (see section 2.2.3). As an alternative one can think about other ways of generating indistinguishable N-tuples of photons [TBvA09]. In the case of multiple frequencies, or in the broadband limit, other issues (such as limited vsisibility) might arise [KY02, CW04]. Also one has to take into account how photon losses can affect the scheme [GAW+10]. Also the question about how to generate arbitrary patterns has to be addressed [BSSS01a, BSSS01b, DSS06, KBA+01].

4.3 Eﬃciency

An important question which has not been addressed much so far is how efficiently one can make quantum lithography or quantum imaging. By efficiency I mean mainly the exposure time aspect. Let’s for example imagine that one wants to use quantum lithography for making an image of an object with higher resolution than in the classical case. The question is how long time is needed to get such a 46 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY pixellated image. Some objects might be stationary for very long times and the importance of the sample might be so high that one has no problem to wait hours or even weeks for an image, but for most practical applications one would not accept to wait for more than, say one hour, or, if the sample is unstable or moving one might not even accept to wait longer than a few minutes or seconds. Similar considerations can be done for lithography schemes. To produce one sample with quantum lithography one might accept that it takes some time, but for doing mass production with hundreds, thousands or even millions of produced objects one will not be able to do it efficiently if one has to wait too long time for every single object to be illuminated. What I investigate in this thesis is the following question, closely related to the above mentioned problem: If one wants to get a specific number of N-photon states onto one pixel on the average, how does the time needed for that to happen scale with the number of pixels and the number of photons in the N-photon state? Keeping constant the number of N-photon states which on the average reach one pixel implies that one keeps the visibility constant. To answer the question, one can imagine an experiment similar to those in section 4.2.2 by looking at Fig. 4.2. A detailed description and analysis of it is given in section 4.3.4 and section 4.3.5, but the important thing is that the N00N- state is present after the double-slit, i.e., one has either N indistinguishable photons of wavelength λ passing through the upper or N indistinguishable photons passing through the lower of the two slits. The photons then propagate from the slits to the detection plane, where they form an interference pattern. The detection plane is placed a distance R after the slit, where this distance is much bigger than the distance d between the double-slit. Here, and from now on, I will consider a one- dimensional detection plane. There are two reasons for that. On one hand it is enough to answer the question above, since it will make no difference in which direction one changes the number of pixels to see the scaling behaviour. On the other hand, as will become clear later on, my implementation of an experiment to answer the question produces already two- or three-dimensional plots. Taking a two-dimensional detection plane would somehow require 4- or 5-dimensional plots and the experiment would take much longer time to perform without gaining any other insights than in the case with a one-dimensional detection plane. A total number of S + 1 detectors is placed at the detector plane, tightly beside each other to cover the whole length of the plane. The width b of the detectors is small enough to allow to resolve the details of the interference pattern. The detectors could simulate pixels in an imaging scheme. The “classical” interference pattern is achieved if one sets N = 1, i.e., if one allows one photon at a time to pass. At the slit one will then have the state

1 |Ψ1,sli = √ |1i ⊗ |0i + |0i ⊗ |1i . (4.4) 2 upper lower upper lower

Since I choose not “to look” through which slit the photon went, one will have the 4.3. EFFICIENCY 47

relative intensity →

 2-photon events ------1-photon events

Figure 4.2: Scheme of an interference experiments: A double-slit with slit distance d is placed on the left. The photons passing through the slit diffract and propagate to the detection plane on the right. In the detection plane detectors of width b are tightly placed beside each other. In case of classical light or when only allowing one photon passing through the slits per coherence time one would get the indicated one-photon interference pattern. If one now keeps the wavelength of the light and all other parameters constant, but only allows pairs of photons to pass with equal probability either through the upper or through the lower slit and one only detects the case where both photons arrive at the same detector one would get the indicated two-photon pattern, i.e., a pattern with half the distance between two peaks. To see the pattern the width of the detectors has to be sufficiently smaller than the width of a peak (not indicated in the figure). To get the specific patterns plotted here, I assumed that the distance between the slits is three times the width of a slit. See the text for more details about the experiment.

state

S 1 X2 n o |Ψ i = eikr(s)aˆ† + eik[r(s)+∆r(s)]aˆ† |0i 1,sc N s s 1 S s=− 2 S 1 X2 h i = eikr(s) 1 + eik∆r(s) aˆ† |0i , (4.5) N s 1 S s=− 2

p at the detection plane, where N1 = 2(S + 1), the summation is√ over all detector modes s, k is the wave vector of the incident light, r (s) = R2 + s2b2 ≈ s2b2 R 1 + 2R2 and ∆r (s) = d sin θ ≈ dθ = dbs/R. [For the normalisation in Eq. (4.5) I have for simplicity assumed that the pattern falling on the S+1 detectors contains exactly an integer number of fringes.] 48 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY

This leads to a probability of detecting a photon at a specific detector as kdbs P (s) ∝ cos2 , (4.6) 1,sc 2R where I omitted the envelope function which is produced by the finite width of a slit. In other words, Eq. (4.6) describes the distance between two peaks for single- photon events in Fig. 4.2, but not the height of each peak. The resulting distance between two peaks is thus λR/d. One should note that this distance is not affected when one takes into account the envelope function. Although it is not much of a question how single photons act in this example, there exist (at least) two different theories explaining what is happening to the photons if N ≥ 2, especially how the photons must propagate between the double- slit and the detection plane. These two theories predict the same practical results for several cases and could thus have equally been used to describe most of the experiments in section 4.2, but they predict quite different exposure time scaling behaviours, the question I addressed above. First, I show the two different theories and their implications on the scaling behaviour. Then I present an experiment which actually gives different results for the two theories before I analyse this experiment.

4.3.1 The theory of Boto et al. According to Boto et al. [BKA+00], the N-photon state will stick together after having passed through the slit, and subsequently all the N photons will arrive at the same detector. His original words for the case N = 2 and the probability for the two-photon state to arrive at a speciﬁc point on the screen depending on the intensity I of the incoming light, are as follows: “For two-photon absorption with entangled photon pairs, the absorption cross section scales as I... If the optical system is aligned properly, the probability of the ﬁrst photon arriving in a small absorptive volume of space time is proportional to I. However, the remaining N −1 photons are constrained to arrive at the same place at the same time, and so each of their arrival probabilities is a constant, independent of I.” Looking at this case N = 2 and implementing it to my experiment the state 1 |Ψ2,sli = √ |2i ⊗ |0i + |0i ⊗ |2i . (4.7) 2 upper lower upper lower at the slit would propagate to the state

S 1 X2 h i |Ψ i = ei2kr(s) 1 + ei2k∆r(s) (ˆa†)2 |0i , (4.8) B,sc N s B S s=− 2 p at the detection plane, where NB = 4(S + 1). At this point I will not address how the state in Eq. (4.7) can be produced, nor how or if it can be made to propagate 4.3. EFFICIENCY 49

in such a manner as to produce the state |ΨB,Sci at the detection plane. This will be discussed later in section 4.4. Comparing Eq. (4.8) with Eq. (4.5) one can see that, like in the concept of photonic de Broglie-waves [JBCY95], the phase has been doubled. This implies that the distance between two peaks at the screen is now only λ0R/(2d), i.e., half as long as the distance given by Eq. (4.6).

4.3.2 The theory of Steuernagel Steuernagel [Ste03, Ste04] opposed the description of Boto et al. by questioning that the photons somehow “stick together” after the slit. He assumes that the two (or, in general, N) photons in the N00N-state should rather propagate independently after having passed the slit. In [Ste04] he writes about the theory of Boto et al. “... it is not true that the ﬁrst arriving photon greatly constrains the arrival location of the following ones ... Very few photons will be absorbed in one point since they typically arrive far apart.” If one transfers his idea into the mathematical picture presented here one would get

S S 1 X2 X2 n o |Ψ i = eik[r1(s)+r1(t)]aˆ†aˆ† + eik[r2(s)+r2(t)]aˆ†aˆ† |0i(4.9) St,sc N s t s t St S S s=− 2 t=− 2 at the detection plane√ when the state in Eq. (4.7) propagates. Here I introduced the notations NSt = 2(S + 1) and

s2b2 db r (s) = R 1 + ∓ s. (4.10) 1,2 2R2 2R r1,2 are the distances between the place of detection and the upper or lower slit, respectively. Since the photons propagate independently they typically hit two diﬀerent detectors. Therefore, I introduced a second detector variable t in the above equations. Although Eq. (4.9) might seem intuitively clear, I will show a thorough theoretical analysis starting from the pump beam to the detector in section 4.3.5.1 which at the end leads to Eq. (4.9). When only looking at two-photon detections at the same detector (i.e., s = t), one will get the same distance λ0R/(2d) between two peaks as in the description of Boto et al., i.e., half as long as the distance given by Eq. (4.6).

4.3.3 Comparison of the theories of Steuernagel and Boto et al. As mentioned above the interference pattern observed by looking at two-photon detection at the same detector is the same when applying the theory of Boto et al. or when applying the theory of Steuernagel. However, in the ﬁrst case all photons arrive always at the same place, and thus every N00N-state passing the double-slit will contribute to the interference pattern, whereas this is not the case in Steuer- nagel’s description. In the latter description very few of the N00N-states passing 50 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY through the slit will hit coincidently with both photons the same detector and thus contribute to the interference pattern. Mathematically this can be expressed by the two relations for the detection probabilities

kdbs P (s, t) ∝ cos2 δ (4.11) B,sc R st kdb P (s, t) ∝ cos2 (s + t) , (4.12) St,sc 2R where δst denotes the Kronecker delta. The probabilities will decrease the more pixels (smaller detectors) one uses, since this will lower the probability of hitting the same detector. Even when going from N = 2 to higher N (and therefore to lower distances between two peaks, i.e., one gets a higher resolution) the probability of arriving at the same detector will be lower in Steuernagel’s desription, whereas it will stay constant in the description of Boto et al. If one calls these “exposure times” τ, i.e., the time one has to wait on the average to get a certain number of N-photon “hits” on one of the S detectors, into a scaling law they will scale as

τB(S, N) ∝ S (4.13) N τSt(S, N) ∝ S . (4.14)

The implication of these different scaling behaviours is quite dramatic. One can assume the simplest case of N = 2 and that one wants to illuminate a 100 × 100 pixel sized image. One also assume that the absolute prefactor of the exposure time is the same in both cases. This is justified by the fact that the difference in the two theories is only how the photons propagate. Anything else affecting the exposure time, like detection efficiency, attenuation in air or fibres or other effects, will be the same in both descriptions. Therefore, comparing the exposure times of Steuernagel and Boto et al. one gets in the above example that it will be 10,000 times higher if Steuernagel description is right compared to the description of Boto et al. Doubling the number of pixels in each direction, so that one will have a picture of 200 × 200 pixels, the descriptions differ already by a factor of 40,000. Staying at 100×100 pixels but getting a better resolution by adding one photon in the N00N- states so that one has N = 3 takes 10,000 times as long as in the case of N = 2 in Steuernagel’s description and the difference in exposure time between Steuernagel’s description and the description by Boto et al. will be already 108. Assuming that one is able to produce enough N = 2 or N = 3 N00N-states per second that one has on the average a statistical error of 10 per cent at each pixel and that the intensity is equally distributed, one needs to produce 1 million N00N-states per second if one would have no photon losses and a detecting efficiency of 1 and one wants not wait more than 1 second before the image is given in the description of Boto et al. Since most of the single-photon detectors (SPDs) commercially available today have a maximum gating frequency in the order of MHz this is at the forefront of what is theoretically possible to implement today (although far away from implementing 4.3. EFFICIENCY 51

Figure 4.3: Schematic experiment for measuring spatial correlations. This is a practically more suitable version of the experiment in Fig. 4.2. The coordinates of the detectors are denoted s and t. The nonlinear crystal is placed directly in front of the double-slit to assure that only N00N-states pass through it. See the text for more details. in practice, yet). However, taking this optimistic case and using the Steuernagel’s description instead one would have to wait around 3 hours for one imaging process if N = 2 and around 3 years (!) if N = 3. Unfortunately, as it will turn out, Steuernagel’s description seems to be the correct one. For making lithography of any practically useful sample one has, in my opinion, rather to have even a higher number of pixels, which makes me to question most of the practical use one could have dreamt on in quantum imaging or quantum lithography schemes.

4.3.4 Conclusive experiment As already indicated above one could perform an experiment which gives different outcomes for the two theories by looking at correlations of detectors at different locations. I will first present the idea of such an experiment and then its practical implementation by Peeters et al.

4.3.4.1 Idea of the experiment There are two problems when directly performing an experiment as depicted in Fig. 4.2. Firstly, SPDs are very expensive, making it basically impossible to build an array of a large enough number of detectors. Secondly, the detectors should dis- criminate 1-photon events from two-photon events. Those photon number-resolving detectors (PNRDs) are very hard to build and thus an array of PNRDs is even more utopian to realise today. Instead one can use a setup as depicted in Fig. 4.3. Instead of using PNRDs, one can split the beam at a 50:50BS before it enters the detection plane and instead of using a lot of detectors to cover the whole detection plane only two movable detectors are used as indicated in the ﬁgure. The two detectors are then connected to a coincidence unit which gives a signal if both detectors detect a 52 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY

(a) (b) Figure 4.4: Diﬀerence between the two theoretical explanations for N = 2 when measuring the coincidences. In (a) the model of Boto et al. is displayed as given in Eq. (4.11). In (b) the model of Steuernagel is illustrated as given in Eq. (4.12). The axes correspond to the positions s and t of the two detectors in Fig. 4.3 respectively. The distance between two peaks along the line s = t is half as long as it would be for classical light or for the N00N-state consisting of only one photon. The lighter the colour, the higher the coincidence rate is. The colour schemes in the two ﬁgures are only relative. One can see that for s = t the two theoretical models give the same pattern.

photon during the same time window. By moving the two detectors independently one is able to detect all possible correlations between the detectors as they would be placed in the array of Fig. 4.2. Setting the coordinate of one detector as the abscissa and the other one as the ordinate one can plot the possible correlations in the experiment according to the theory of Steuernagel or the theory of Boto et al. as given in Eq. (4.11) and Eq. (4.12). Such a plot is given in Fig. 4.4. As one can see, the two theories discussed above give very diﬀerent outcomes for such an experiment and therefore, by performing it, one could tell which of the theories is applicable to describe quantum lithography (or at least which one is not). One should note that the colouring scheme in the ﬁgure is only relative. Since all two-photon states should arrive somewhere at the detection plane the absolute intensity at one point for s = t would otherwise be much higher when applying the theory of Boto et al. compared to Steuernagel’s theory. One should also note that I have assumed that the relation between the slit width, slit distance, and the wavelength is such that one can see all these peaks. For less advantegeous values one could see less peaks, as indicated in Fig. 4.2, where in that case only 3 or 5 peaks would be easily visible. 4.3. EFFICIENCY 53

4.3.4.2 Realisation of the experiment by Peeters et al. Peeters et al. [PRv09] built an experiment similar to that depicted in Fig. 4.3. In their experiment they allowed the state

Peeters c1 Ψ = √ (|2i ⊗ |0i + |0i ⊗ |2i ) + c2 |1i ⊗ |1i , (4.15) 2,sl 2 u l u l u l where u indicates the upper slit, l indicates the lower slit, ci ∈ C, i ∈ {1, 2} and 2 2 |c1| + |c2| = 1, to pass through the double-slit. Since the parameters ci could be chosen freely, one sees that Eq. (4.7) is a special realisation of Eq. (4.15) with |c1| = 1 and c2 = 0. In order to have the freedom to choose these parameters, they did not place the non-linear crystal directly in front of the double-slit, but used some Peeters lenses between the crystal and the slits to be able to produce the Ψ2,sl -state. The important thing, however, is that they were able to produce the state of Eq. (4.7) experimentally and their result for that case is printed in Fig. 5 (a) of their paper [PRv09]. This ﬁgure has the same pattern as Fig. 4.4 (b) of this thesis, which shows that the description of Boto et al. about how photons arrive at the detector is wrong, at least if the photons propagate freely between the slits and the detector planes, and indicates that Steuernagel’s description seems to be the correct one.

4.3.5 Theoretical analysis 4.3.5.1 Analysis of the experiment of Peeters et al. A thorough theoretical analysis of the experiment of Peeters et al. shows, that the state produced for |c1| = 1, c2 = 0 in Eq. (4.15) indeed leads to the state of Eq. (4.9) at the slit. Such an analysis is done in Appendix A in paper D, which is part of this thesis. The important thing from that analysis is that the state of Eq. (4.15) is not due to some speciﬁc parameters of the crystal like width or length (except the fact that degenerated, collinear, type-I conversion is used to make the down-converted photons indistinguishable in their polarisation and wavelength), but that this is a rather general expression not opening for any backdoors to rein- troduce a behaviour as in the description of Boto et al. by only changing some parameters. Other interesting results which one can conclude from the experiment of Peeters et al. and my analysis thereof, but which is not written in their paper [PRv09], arise if one looks at other values ci in Eq. (4.15) than those leeding to the state iϕ in Eq. (4.7). To this end, I write c1 = e sin(α/2) and c2 = cos(α/2). The plots for some values of ϕ and α are given in Fig. 4.5. When comparing them with the experimental results in Fig. 5 of the paper by Peeters et al., one can see a very good agreement between the experiment and the theory. Going from the lower left corner to the upper right corner along the diagonal line in Fig. 4.5 (a), which corresponds to the case of measuring the interference pattern as in Fig. 4.2 and which corresponds to the N00N-state, one can count 9 fringes. 54 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY

(a) (b) (c)

(d) (e) (f) Figure 4.5: Diﬀerent theoretical plots of Eq. (4.15) where each axis represents one detector (as in Fig. 4.4). A lighter colour indicates a higher intensity. The parameters are (a) α = π, ϕ = 0, (b) α = 3π/2, ϕ = 0, (c) α = π/2, ϕ = π/2, (d) α = 0, ϕ = 0, (e) α = π/2, ϕ = 0, and (f) α = π/4, ϕ = 0. These calculated plots should be compared to the experimental results of Peeters et al., presented in Fig. 5 of their paper [PRv09].

Looking at Fig. 4.5 (e) instead and counting fringes in the same manner, one ends up at only half as many fringes, i.e., one has only half the resolution compared to the previous case. The situation in Fig. 4.5 (e) corresponds to what one would expect using a setup as in Fig. 4.3 with classical light of the same wavelength. This comes not as a surprise since the state from Eq. (4.15) represented by this figure is the most classical one. Fig. 4.5 (b) and Fig. 4.5 (f) are mixtures between these two extremes and therefore some of the peaks are not as high as the other ones. A difference between the most “classical” case and the N00N-state is what happens if one goes along the diagonal line and has both detectors at a place where zero coincidences are detected, i.e., a place where two photons never arive at the same place. In the “classical” case, keeping one detector fixed at that place and moving the other one (corresponding to going up and down, or to go left and right in the figure), one would never see any two-photon coincidences between these two detectors. In the case of the N00N-state, however, redoing the same, one suddenly 4.4. IMPLEMENTATION OF THE CONCLUDING EXPERIMENT IN PRACTICE 55 sees coincidences when moving one detector and keeping the other one in place. This shows that in the case of the two-photon N00N-state, one can have one- photon detection at some locations on the screen in Fig. 4.2, but never two-poton detection at the same location. In the “classical” case, however, no two-photon detection excludes also single-photon detection at that location. An analysis of the other plots in Fig. 4.5 is given as well in Appendix A in paper D.

4.3.5.2 Analysis of the original experimental proposal by Boto et al. An important question is whether my main results from section 4.3.4 can be trans- formed to the original experimental proposal of Boto et al. [BKA+00]? After all, they proposed two-photon interference between two plane-wave like, e.g., Gaus- sian beams. The answer is yes. The photons have to have some origin like a laser or a non-linear crystal and are in a diverging mode, even if they are, for example, produced at “exactly” the same place via spontaneous parametric down- conversion (SPDC) in a crystal. A thorough analysis of how the N00N-states propagate to the detector plane, even under diﬀerent initial conditions, is given in Appendix B of my paper D, which is included in this thesis. There, it is shown that the photons will not “stick together” in a strict sense but will slowly move away from each other and when hitting the screen the situation will be as in the experiment of Fig. 4.2, i.e., the photons have very low probability to hit the same point, but will, in general, appear at diﬀerent places of the interference pattern. Therefore one has the same scaling behaviour as in Eq. (4.14), instead of Eq. (4.13).

4.4 Implementation of the concluding experiment in practice

To implement the experiment in Fig. 4.3 and to assure that only the N00N-state with N = 2 is present, Peeters et al. used continuous wave (CW)-lasers in their experiment. To get the N00N-state for higher numbers of N it seems, however, that pulsed lasers are needed, since ordinary tabletop CW-lasers are not able to achieve high enough powers to generate photons of higher order nonlinearites at reasonable rates in the nonlinear crystals. In this section I show what one has to look at when choosing a suitable crystal for doing the setup with pulsed lasers and then what problems might arise with the slit and how they can be avoided.

4.4.1 Crystal Beside the quite obvious demand that the nonlinear crystal should produce down- converted photons at a fairly high rate (an overview about most of the crystals one can use and its material characteristics is given in [DGN99]), one has to chose the length of the crystal carefully. One has to fulfil two contradicitive constraints. On one hand, the crystal should be as short as possible. The reason is that, although the crystal is optimised for collinear phase-matching, the pump beam is always 56 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY diverging. Assuming the pump beam to be Gaussian and that the minimum waist of the Gaussian is in the middle of the crystal, one will get pump light enclosing an angle greater than zero with the optical axis before and after the middle of the crystal and these diverging parts of the pump have, due to the geometry of the crystal, slightly different phase-matching conditions to fulfil. These different phase- matching conditions might allow the down-converted photons to be emitted in a cone instead of being emitted collinearly and one thus has to assure that the opening angles of the cones are so small that the photons from the point of generation (in the worst case the beginning of the crystal) until the place where they hit the slit have no possibility to pass through different slits, since this would result in that one no longer has the N00N-state after the crystal. On the other hand, the crystal should be as long as possible [BBC+00, GMSW02, EBGW00], since the phase- matching function Φ is proportional to sinc(∆kL/2), where L is the crystal length, ∆k = kp − ks − ki is the phase mismatch and kp,s,i are the wave numbers of the pump, signal and idler photons, respectively. The amount of converted photons is proportional to Φ and thus the shorter L the higher values of ∆k are allowed and the higher deviations from the degenerate case λi = λs = λp/2, with λ being the wavelength of the photon, are possible. Deviations from this degenerate case would make the photons in principle distinguishable, which is not wanted by my setup. Also the width of the pump beam has to be taken into account [BBC+00]. A small width assures high intensities in the crystal and makes that many photons pass the slit instead of getting absorbed by the opaque parts. On the other hand any finite width would allow for transverse components of the wave vectors of signal and idler and thus making it possible for these photons to be emitted in a non-collinear way and therefore pass different slits. Finally, one has to be aware of the effect that, even in the collinear case, not only degenerated photons (i.e., the two down-converted photons have the same wavelengths) are produced, but also photons at other wavelengths. This is mainly due to short pulse-length of the pump, so that it consists of a range of different wavelength. Therefore one has to solve the phase-matching conditions for all these wavelengths and this results in many different wavelengths at the signal and idler photons as well. A detailed investigation is given in the papers [BK08, BK09]. The bandwith of these produced photons is around 100 nm which makes it unavoidable to use filters to narrow the bandwith drastically, since the interference pattern would be smeared out otherwise. This considerable lowers the percentage of down- converted photons passing the filter.

4.4.2 Slit Since the slit has to be placed very close to the crystal, there is no chance to filter away the not-converted pump beam before the slit. The pump beam has, however, a quite high peak power. As I used a titanium-sapphire (Ti:S)-laser with a pulse rate of around 80 MHz and pulse lengths of around 60 fs with a central wavelength of λ = 780 nm which is first converted to 390 nm with an average output power 4.5. OUTLOOK AND OPEN QUESTIONS 57 of the blue light of 1.2 W, one can calculate the peak power to 250 kW, assuming for simplicity equally distribution of the power during the time of the pulse and no change in pulse duration after the conversion. Taking into account the width of the pump at the slit, one has to assure that this energy can be absorbed by the opaque parts of the slit without damaging this material or heating it up so much that it will expand too much. Since the slit has to be quite thin to not affect the beam too much, it is usually put on a transparent substrate to make the slit stable despite its small thickness. This transparent substrate, on the other hand, should not be fluorescent at the high powers of the beam since this fluorescence could emit in the same wavelength region as the down- converted photons. At the same time the details of the slit have to be quite small, where I chosed slitwidths of around 100 nm and a slit distance of around 200 nm. It turned out that only gold worked amongst the tried materials. In other materials like ink, nickel or copper the intense light burned holes in it, making them unusable as slits. As the substrate I used fused silica, since other glasses usually showed some of the mentioned unwanted fluorescence effects at the wavelengths I was using. Since the gold is not sticking to the glass directly, one has to put a small 4 nm thick titanium layer between the glass and the gold since the titanium is sticking both to the glass and to the gold. On top of that I put a 600 nm thick gold layer. This turned out to be thick enough to absorb the energy from the pump without destroying the layer. Finally, I put a photo-resist on the gold layer and a mask on top of it. The mask could be printed on a transparent sheet via a high resolving office copy machine. After shining with UV-light on the mask one could remove the mask, develop the photo-resist, etch away the gold from the places where there was no mask and finally remove the remaining resist. The titanium layer will remain, but an investigation under the microscope shows that it is still transparent for light of 780 nm, although some photons might be absorbed. The whole process of producing the slit is depicted in Fig. 4.6.

4.5 Outlook and open questions

I have shown that the original assumption of Boto et al. about how quantum lithography works was too optimistic and that it seems difficult to find “real life” applications of quantum imaging or lithography in applications where time is an issue. On the other hand, one might wonder if there are any ways to get around the unfavourable time-scaling behaviour. I will now first look at whether or not one can get the needed correlations for a better scaling behaviour by changing or rearranging any parts of the setup, then I will comment on some papers who claim that spatial correlation can be retained, and finally I shall discuss what else one can do. 58 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY

Figure 4.6: Schematic sketch about how to produce the slits. See the text for more details.

4.5.1 Are there any other possible correlations?

In section 4.3.5 I did two assumptions which should be motivated. First, I assumed that the thickness of the slit is thin compared to the width of the slit, so that I do not have to use an extra propagator for the time where the light passes the slit but only for the time before and after. To my knowledge, no one has ever proposed an experiment in the domain of quantum lithography or imaging where this is not fullfilled. I can not see any reason how a thicker slit would give new correlations to the diffracted beam. The second assumption is that the slit width is smaller than any spacial structure of the beam profile. What happens if this condition is not fullfilled? One can look at this situation by taking into account the slit width a as well. In the thin crystal approximation with perfect phase matching such that signal and idler are always at the same wavelength (this can be easily assured by placing narrow-band filters in front of the detector) one has total anti-correlation between 4.5. OUTLOOK AND OPEN QUESTIONS 59 the transverse momenta (see for example [MSP98] for thin-crystal approximation), 0 0 ˜ 0 0 † † ˜ 0 0 i.e., one has the wave function |k1, k2i ∝ Ψ(k1, k2)â 0 aˆ 0 |0, 0i with Ψ(k1, k2) ∝ k1 k2 0 0 ~ δ(k1 + k2), where I restrict myself only to one component of the wave vector k, namely the one perpendicular to the slit opening lengths. This is motivated by the fact that the interference pattern will appear in that direction and that the detectors are moved only in that direction. If one does not have a thin crystal but still perfect phase-matching, one can write the state as

˜ 0 0 X 0 0 Ψ(k1, k2) ∝ f(∆k)δ(∆k + k1 + k2), (4.16) ∆k where the sum is over all possible linear momentum oﬀsets ∆k and f is a weighing function. The general case is impossible to solve, but for my purpose it is suﬃcient to look at the case where one has

˜ 0 0 0 0 0 0 Ψ(k1, k2) ∝ δ(k1 + k2) + δ(∆k + k1 + k2) + δ(−∆k + k1 + k2). (4.17)

I will motivate it later on. If one makes a Fourier transform of Eq. (4.17) to get the ﬁeld proﬁle in coordinate space one gets, after some algebra,

Ψ(x1, x2) ∝ δ(x2 − x1) [1 + 2 cos(∆kx1)] . (4.18)

After the double slit, with a and d deﬁned as before, one gets

Ψslit(x1, x2) ∝ δ(x2 − x1) [1 + 2 cos(∆kx1)] (4.19) 2 Y xn − d xn + d × u + u , (4.20) a a i=1 where u is the rectangular function

1 if |x| ≤ 1/2 u(x) = . (4.21) 0 if |x| > 1/2

Since one has to demand that the photons pass through the same slit to make the scheme work (i.e., that one has the N00N-state ), one can write Eq. (4.20) as

Ψslit(x1, x2) ∝ δ(x2 − x1) [1 + 2 cos(∆kx1)] (4.22) x + x − 2d x + x + 2d × u 1 2 + u 1 2 . (4.23) 2a 2a

Transferring this expression back to momentum space gives, after some algebra,

1 ˜ X Ψslit(k1, k2) ∝ sinc [a(k1 + k2 + m∆k)] cos [d(k1 + k2 + m∆k)] , (4.24) m=−1 60 CHAPTER 4. QUANTUM IMAGING AND LITHOGRAPHY where sinc(x) = sin(x)/x. If one places the detectors in the far-field such that one has Fraunhofer diffraction, one can define the angles ϑ1,2 = k1,2/k as coordinates for the detectors and write the previous equation in these coordinates to get the field profile

1 X Ψslit(ϑ1, ϑ2) ∝ sinc [ak(ϑ1 + ϑ2 + mϑ0)] cos [dk(ϑ1 + ϑ2 + mϑ0)] (4.25) m=−1 at the detectors, where ϑ0 = ∆k/k. For comparison one can take one step back for a moment and look at only one k-mode (i.e., ∆k = 0) and at the case of two-photon detection at the same place (i.e., ϑ1 = ϑ2), which gives us the detection probability

|Ψ(ϑ)|2 ∝ sinc2(2akϑ) cos2(2kdϑ). (4.26)

To calculate the one-photon detection probability one has to calculate the integral R 2 dϑ2|Ψ(ϑ1, ϑ2)| , which is impossible to do analytically. However, a classical treatment of the double slit [BW99] gives that the intensity is proportional to cos2(dkϑ), which is half of the period of the two-photon case as it should be to make quantum lithography working. Going back to Eq. (4.25) one can see that the cosine-function oscillates much faster than the sinc-function, since d by deﬁnition has to be larger than a. Therefore, one can treat the sinc-function as a constant for small values of ϑ1,2 and write Eq. (4.25) after some short algebra as

Ψ(ϑ1, ϑ2) ∝ cos [kd(ϑ1 + ϑ2)] g(ϑ0), (4.27) where g is a function only depending on ϑ0 (which itself only depends on ∆k) and not on ϑ1,2. Here, again, nothing indicates that the photons stick together and the result is the Steuernagel-prediction, i.e., the two photons can propagate in different directions before they hit a detector. As one can see from the calculations, any value of ∆k, when assuming three equally weighted beams, gives the same shape of the curve. By orienting the crystal correctly one can always achieve this symmetry. But as long as one has this symmetry it does not matter whether one has 3 or any other number of modes and whether these modes have different amplitudes. One will always end up at Eq. (4.27). Only g, i.e., the overall intensity will change. Other correlations might be achieved by allowing different wavelengths for signal and idler. However, this will make the photons distinguishable and therefore break a prerequisite for quantum imaging schemes. In conclusion, a theory including any type of non-linear crystals shows that there seems to be no possibility to make the photons stick together as in the proposal of Boto et al. in a linear propagation medium. 4.5. OUTLOOK AND OPEN QUESTIONS 61

4.5.2 Other investigations about linear scaling behaviours In literature there are some schemes which show that a scaling behaviour as claimed by Boto et al., i.e., the intensity for N-photon detection is direct proportional to the intensity of the detection of one photon, might be possible. Two such papers are [JG90] and [PST98]. They assume, however, either explicitly or implicitly, that the photon somehow arrive at the same place to have such a relation between the intensities without stating how this would be achieved. This is, however, not the case in quantum lithography, as I have shown, since the photons diﬀract independently and thus do typically not arrive at the same place. In order for these proposals to be credible, one would require an explicit mechanism for the assumed spatial correlation of the photons.

4.5.3 Using other media Another way to keep the photons together after the slit might be to use a nonlinear medium between the slits and the detectors. In that case, one can’t use air or vacuum any longer, but might achieve something like spatial solitons [Kel64, BMF85]. This would have to be investigated further, but it seems to be quite farfetched to eﬃciently implement such a scheme with only very few photons.

4.5.4 Outlook To facilitate a large-scale use of quantum imaging or quantum lithography even with a small number of N is not possible with the proposals given so far. Although these setups work in principle, their scaling behaviour as given in section 4.3.3 prevents most of its practical use. It seems therefore as a main challenge for the future to overcome these restraints when doing further investigations in the ﬁeld of quantum imaging or lithography.

Chapter 5

Measurement of entanglement

5.1 Introduction

To determine the entanglement of an unknown state ρˆ quantitatively can be quite cumbersome and time-consuming. The reason is that one, in general, has to know the whole density matrix of ρˆ to determine its concurrence (see section 2.3.1) or any other measure of entanglement. The density matrix can be determined by doing state tomography of ρˆ [Leo96, LR09]. Full state tomography of two qubits requires already nine measurement settings and since the number of measurement settings is proportional to the number of entries of the density matrix this is hard to do in reasonable time for higher-dimensional systems. Another problem of state tomography is that the measurements, in general, have to be performed in a common reference system. Especially when the qubits (or higher-dimensional systems) are measured at different locations it may be difficult to align the respective reference frames. I tried to find a method for quantifying entanglement which is experimentally easier to implement than state tomography, i.e., which requires fewer measurement settings and is invariant under local unitary transformations, so that a common reference system is not needed any longer. I will first present such a method and then discuss it for pure states and mixed states of two qubits before I discuss generalisations for higher-dimensional systems and finally give an outlook and discuss open questions. More details and the background about how I arrived at the presented method can be found in the paper A, which is part of this thesis. It was pointed out after the publication of my work that my definition in Eq. (5.2) was already discovered in [SM95]. The authors of that work could, however, not relate their results to the concurrence C as described below, since the concept of concurrence didn’t exist at that time.

63 64 CHAPTER 5. MEASUREMENT OF ENTANGLEMENT

5.2 Quantifying entanglement through correlations of local observables

As is argumented in paper A the information of entanglement between two systems A and B should be somehow coded in the covariance

D E D ED E C Aî, Bî = AîBî − Aî Bî , (5.1)

n o n o where Aî and Bî are sets of observables acting only on the respective subsystem and having no joint eigenvector. With this definition one can define a measure G2Q as the sum of all possible covariances between two local sets of mutually unbiased observables which span the whole system:

3 X 2 A B G2Q = C σˆi , σˆj , (5.2) i,j=1

A B where σˆi denotes the i:th Pauli matrix [see Eq. (2.32)] for system A and σˆj the j:th Pauli matrix for system B. This measure is invariant under any local unitary transformation or under partial transposition. Therefore, no common reference system is needed and since

A B A B A 1ˆB 1Â B C σî , σˆj = σî ⊗ σî − σî ⊗ ⊗ σî , (5.3) only single rates and coincidences have to be counted. For a two qubit system one has to measure at 9 different settings (the results where one of the operators on the subsystems is 1ˆ can be calculated from the other results). For this case there is no reduction in the number of measurements compared to full state tomography yet, but the simple locally implementable measurement methods make G2Q a good experimentally implementable measure. One should note that the measure can also be written as

h 2i G2Q = 4Tr (ˆρ − ρˆA ⊗ ρˆB) , (5.4)

where ρˆA denotes the density matrix after tracing over subsystem B and vice versa for ρˆB. It is also not necessary to use the Pauli matrices in the deﬁnition of Eq. (5.2). Any other local unitary transformation of this mutually unbiased basis (MUB) [DEBŻ10] works equally well. 5.2. QUANTIFYING ENTANGLEMENT THROUGH CORRELATIONS OF LOCAL OBSERVABLES 65 5.2.1 Pure states of two qubits Linden and Popescu [LP98] showed that there are two invariants for two qubits under local unitary transformations:

1 X ∗ I1 = aklakl, (5.5) k,l=0 1 X ∗ ∗ I2 = akmaknalnalm, (5.6) k,l,m,n=0 where a is deﬁned as in Eq. (2.34). One sees that I1 = 1 is nothing but the normalisation condition and in paper A I show that G2Q can be written as a function 2 2 of these invariants, namely G2Q = 8Iβ + 16Iβ, where Iβ = (I1 − I2)/2. It turns 2 also out that Iβ can be written as Iβ = C /4 [AF01], where C is the concurrence as deﬁned in section 2.3.1, and therefore

2 2 G2Q = C 2 + C (5.7) for pure states. Since the concurrence cannot be negative there is a one-to-one correspondence between C and G2Q for pure states.

5.2.2 Mixed states of two qubits For mixed states of two qubits there is no longer a one-to-one correspondence between G2Q and C, but instead one has

2 2 2 C 2 + C ≤ G2Q ≤ 1 + 2C (5.8) and therefore 0 ≤ G2Q ≤ 3. A motivation for Eq. (5.8) is given in paper A. But G2Q can still be a useful measure even for the case of mixed states. Since C > 0 implies entanglement one can use G2Q as an entanglement witness [HHHH09] if G2Q > 1. Furthermore, if one looks at Fig. 5.1, one can see that high values of G2Q restrict the possible values of C to a narrow range. In the ﬁgure the case G2Q = 2.5 is indicated by a dotted line. For this value one can see that C is somewhere close to 0.9. Compared to entanglement witnesses which can only diﬀer between “entangled” and “undecidable”, G2Q also allows to quantify entanglement within a certain range. One may wonder whether G2Q can be somehow compensated for its “mixedness” so that it becomes a bijective map of entanglement even for mixed states. Simulations given in my paper paper A show that this is not the case when trying to use the purity Tr ρˆ2 [MJWK01] as the compensating factor, but other new measures of “mixedness” might allow such a compensation. 66 CHAPTER 5. MEASUREMENT OF ENTANGLEMENT

2.5

2Q 1.5 G

0.5

0 0 0.2 0.4 0.6 0.8 1 Concurrence

Figure 5.1: Relation between G2Q and the concurrence C. The grey area shows the possible combinations. All the values above the dashed line indicate entanglement. If one detects a high value of G2Q, for example G2Q = 2.5, the possible values for the concurrence are restricted to a narrow range (see the dotted line).

5.2.3 Higher-dimensional systems

G2Q can be generalised to bipartite systems with subsystems A and B of any dimension NA and NB, respectively, by deﬁning

N 2 −1 N 2 −1 A B 2 X X Â ˆB G = C λk , λl , (5.9) k=1 l=1 n o n o Â ˆB where λk and λl are the generators of respective algebras su(NA) and su(NB), where one demands that the generators are traceless and that the trace of the product of any two generators from the same subgroup is zero, except if one has the product of the same generators, where it is 1. This definition of G was introduced in [SB07], [SKB07], and in paper B, which is part of this thesis. The generators can be constructed explicitly [HE81]. Still, as before, the relation h 2i G ∝ Tr (ˆρ − ρÂ ⊗ ρˆB) holds. The maximum value of G as defined in Eq. (5.9) 5.3. OUTLOOK AND OPEN QUESTIONS 67

2 2 is (NA − 1)/NA, if one assumes, without loss of generality, that NA ≤ NB. One has, in general, NA − 1 invariants under local unitary transformations and with some algebra it turns out that for pure states

4 2 Gpure = CI + CI − 6C3, (5.10) where

NA 2 h A2i X 2 CI = 1 − Tr ρˆ = 1 − ai (5.11) i=1 is the I-concurrence as introduced in [RBC+01] and

N XA C3 = aiajak (5.12) i,j,k=1 i 1/4 assures entanglement (due to a different definition of the generators of the algebra compared to the two qubit-case this is different than the relation for G2Q.) A motivation of these results are given in [SB07] and [SKB07]. However, even here states with lower G might also be entangled and so far there is no upper bound for G, making it somehow a quantitative measurement of entanglement. On the other hand the situation is more complicated for systems of higher dimensions than two qubits, since there exist different classes of entanglement and one cannot, in general, transfer a state which has a given value of one scalar invariant under local unitary transformations into another state with the same scalar invariant by using only local operations and classical communications (LOCC). For two qubits this is possible. To quantify entanglement for higher dimensional systems one therefore needs more than one scalar.

5.3 Outlook and open questions

Although I found a measure which is quite easy to implement locally and which is able to quantify entanglement for two pure qubits, it cannot quantify entanglement for mixed states exactly. It would be advantageous if one would ﬁnd a way to compensate the measure for the mixedness of a state which allows a bijective map between C and G2Q even for mixed states of two qubits. Another open question is how to implement “good” measures (i.e., measures which are easily to implement in the lab, which do not need a common reference frame, which need as few settings as possible, and which give an operationally meaningful result). On the other hand, it is still an open question how to best quantitatively describe entanglement for higher dimensional systems.

Chapter 6

Summary and conclusion

In my thesis I looked at correlations in several areas of quantum optics. In the case of phase super-resolution I could show that no quantum mechanical correlations are needed to achieve it, neither direct through entanglement, nor indirect through time-reversal symmetry. There is also no need for joint detection; instead, phase super-resolution can be seen as a solely classical effect. For quantum imaging and quantum lithography I could give experimental evi- dence that the correlations of the photons as proposed by Boto et al. do not exist. As a consequence, it turned out that the efficiency of quantum lithography and quantum imaging is poor. For the entanglement of bipartite subsystems I suggested a measure G which is based on the correlations between the two subsystems. It turned out that the measure is a bijective map of the concurrence for pure states of two qubits and can quantify entanglement in some range for mixed states of two qubits. Correlations are a tricky thing. One has to be careful, since not everything which is quantum mechanical at first sight remains quantum mechanical at the end of the day. However, there are correlations which are solely quantum mechanical and are not allowed by classical physics. To investigate correlations in more areas of quantum optics could be a promising mission for further experiments or theoretical considerations.

Appendix A

Background in mathematics

A.1 Linear algebra

This section gives the deﬁnition of vector and Hilbert spaces which are needed for the axioms of quantum mechanics in section B.1.

A.1.1 Vector space A non-empty set over a field F is called a vector space V if the addition of its elements and the multiplication of its elements with elements of F are defined, such that ∀x, y ∈ V : ∃z = x + y ∈ V and ∀x ∈ V, α ∈ F : ∃αx ∈ V and the following properties, also called vector space axioms, apply for all elements x, y, z ∈ V and for all α, β ∈ F: V1: x + y = y + x, V2: x + (y + z) = (x + y) + z, V3: ∃0 ∈ V : x + 0 = x, V4: α(βx) = (αβ)x, V5: α(x + y) = αx + αy, V6: (α + β)x = αx + βx, V7: 1x = x. In this work I will always use C or Cn with n ∈ N as the field F.

A.1.2 Hilbert space

If one deﬁnes a scalar product (x, y) ∈ F for x, y ∈ V and a norm kxk on a vector space, this is called a Hilbert space H. For a scalar product the axioms are

71 72 APPENDIX A. BACKGROUND IN MATHEMATICS

S1: (x, x) ≥ 0 and (x, x) = 0 ⇔ x = 0, P P S2: x, αiyi = αi (x, yi), i i S3: (x, y) = (y, x)∗. The norm is usually deﬁned as

kxk = p(x, x).

If one has two elements x, y ∈ H, they are called orthogonal if (x, y) = 0. If, in addition, they are normalised, that is kxk = kyk = 1, they are called orthonormal. In this work I will mainly use the Dirac notation, that is |xi ∈ H and hx|xi as the scalar product. If one has diﬀerent Hilbert spaces Hi, 1 ≤ i ≤ n, one can construct a new Hilbert space by taking the tensor product of the Hilbert spaces, n N that is H = Hi. i=1 Appendix B

Basics of quantum mechanics

B.1 The axioms of quantum mechanics

Quantum mechanics is based on ﬁve axioms or postulates:

Axiom 1: Every (pure) state can be described by a normalised state vector |ψi. This vector is an element of a Hilbert space H (see Appendix A for a deﬁnition of Hilbert spaces), called the state space.

The state space of a composite system is the tensor product of its components. If one has n state spaces with state vectors |ψii , 1 ≤ i ≤ n, then the total state vector is given by |ψi = |ψ1i ⊗ |ψ2i ⊗ ... ⊗ |ψni. In this work I will often omit the tensor product sign and use the short hand notations |ψi = |ψ1i |ψ2i ... |ψni or |ψi = |ψ1, ψ2, . . . , ψni instead. Axiom 2: Observables A are described by hermitian1 operators.

Axiom 3: Mean values of observables A are given by hAi = hψ| A |ψi.

The variance of an observable can then be given as

δA2 = A2 − hAi2 . (B.1)

If B and C are two operators, then one can deﬁne the commutator [B,C] as

[B,C] = BC − CB (B.2) and the anti-commutator {B,C} as

{B,C} = BC + CB. (B.3)

1 † ∗ Hermitian means A = A , that is aij = aji, where the asterisk denotes complex conjugation of the matrix entry and A† is called the adjoint of A.

73 74 APPENDIX B. BASICS OF QUANTUM MECHANICS

Axiom 4: If a measurement of the observable A results in the (non-degenerate) outcome ak, the system is immediately transfered into |ki.

The state vectors |1i , |2i ,..., |ni are the eigenvectors of A and the outcomes a1, a2, . . . , an are the eigenvalues to the corresponding eigenvector of the observable A. In case of degenerate eigenvalues ak1 , ak2 , . . . , akm = ak with eigenvectors m P |ak1 i , |ak2 i ,..., |akm i the system will afterwards be in the state |aki i haki |ψi. i=1

Axiom 5: The time evolution of a state |ψi in a closed quantum system is described by the so-called Schrödinger equation

∂ i |ψi = H |ψi . ~∂t The operator H is called the Hamiltonian of the system. The state in a closed 2 system at time t2 can be related to the state at time t1 by an unitary operator U :

|ψt2 i = U (t1, t2) |ψt1 i .

These axioms can be reformulated in terms of density matrices (see section B.3).

B.2 Measurements

If one has a quantum system in the state X |ψi = αi |ψii , (B.4) i where {|ψii} denotes a basis of the corresponding Hilbert space and αi ∈ C, then one can determine the probability to obtain measurement outcomes {an} of an observable A by just taking the absolute square of the scalar product between |ψi and the corresponding eigenvalues {|ni} to the outcomes, that is

2 Pan = | hn|ψi | . (B.5)

B.3 Density matrices

A useful tool in quantum mechanics is the concept of density matrices, also called density operators. With help of the density matrix one can easily determine the diﬀerence between pure and mixed states.

2Unitarity means U †U = UU † = 1, that means, the right and the left inverse of the operator is equal to its adjoint. B.3. DENSITY MATRICES 75

B.3.1 Pure states If one has a quantum system in the state |ψi the density matrix is defined as ρˆ = |ψi hψ|. Let {|ni} be an arbitrary, complete, orthonormal basis. Then one can define the trace Tr as Trˆρ = P hn| ρˆ|ni. The result of the trace operation is always n independent of the chosen basis. By the definition of the trace and the density matrix one can see, that the following equations hold:

ρˆ† =ρ, ˆ (B.6) ρˆ2 =ρ, ˆ (B.7) Trˆρ = 1, (B.8) hAi = Tr(ˆρA). (B.9)

B.3.2 Mixed states

If one has an ensemble {|ψii} of n diﬀerent states, each with the corresponding prob- n n P P ability pi ≥ 0, pi = 1, then one can deﬁne the density matrix ρˆ = pi |ψii hψi|. i=1 i=1 If there is just one pi 6= 0, then one has the situation like in section B.3.1 and such a ρˆ is called a pure state. If more than one pi is non-zero one calls such a state a mixed state. For mixed states one has to change Eq. (B.7) to:

2 2 ρˆ 6=ρ, ˆ Trˆρ < 1, if and only if (iﬀ) pi 6= 0 for more than one i. (B.10) The other equations are still valid. One can use Eq. (B.10) easily to determine if a given density matrix represents a pure or mixed state by just taking the trace of ρˆ2. It should also be stated, that the density operator is a positive operator, that is n X hψ| ρˆ|ψi = hψ| pi |ψii hψi|ψi ≥ 0. (B.11) i=1

B.3.3 Partial trace A very important tool for density matrices is the partial trace. If one has a pure physical system consisting of two subsystems A and B with orthonormal basis states P {|aii} and {|bji} whose state is defined by |ψi = αij |aii |bji and whose density i,j matrix therefore can be written as X X ρÂB = |ψi hψ| = αijαi0j0 |aii |bji hai0 | hbj0 | , (B.12) i,j i0,j0 then one can define

ρˆA = TrB (ˆρAB) (B.13) 76 APPENDIX B. BASICS OF QUANTUM MECHANICS

to be the reduced density operator for system A, where TrB is deﬁned as

X ∗ TrB (ˆρAB) = αijαi0j |aii hai0 | . (B.14) i,i0,j

This deﬁnition gives the right statistics to describe the measurement outcomes. It can be extended straightforward to mixed states as well. In general, the reduced density matrix of a pure state is a mixed state, for example for the four Bell states in Eq. (2.60) - (2.63). Bibliography

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