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Applications and Characterisation of Correlations in Quantum Optics

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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. © Christian Kothe, April 2011 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 mea- sure 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 pro- posals 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. v 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 .
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