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Quantum Nonlinearity Constructed from Individual Photons

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Quantum Nonlinearity Constructed from Individual Photons Faculty of Science Palack´yUniversity Olomouc Quantum nonlinearity constructed from individual photons Mgr. Petr Marek, Ph. D. Habilitation thesis Olomouc 2017 Pˇr´ırodovˇedeck´afakulta Univerzita Palack´eho Olomouc Kvantov´anelinearita budovan´a jednotliv´ymi fotony Mgr. Petr Marek, Ph. D. Habilitaˇcn´ıpr´ace Olomouc 2017 Acknowledgements I would like to express my sincerest gratitude to my colleagues and co- authors, both from the Department of Optics at Palack´yUniversity and from the rest of the world. I would especially like to thank Radim Filip for the countless hours spent discussing topics ranging from mundane to bizarre. Last but not least, I would like to thank my family and my friends. They know what they did. Petr Marek Declaration of originality I hereby confirm that I am the sole author of the submitted thesis which, to the best of my knowledge and belief, contains no material previously published by another person, except where due acknowledgement has been made. The author grants permission to Palack´yUniversity in Olomouc to store and display this thesis and its electronic version in the university library and on the official website. Contents 1 Introduction 1 2 Nuts and bolts of quantum optics 5 2.1 Measurement induced operations . .9 3 Noiseless Amplification 15 3.1 Noise-powered amplification . 18 3.2 Optimal probabilistic measurement of phase . 22 3.3 Summary . 26 4 Probabilistic Quantum Information Processing 27 4.1 Processing with coherent state qubits . 28 4.2 Continuous Information Processing . 33 4.3 Summary . 38 5 Deterministic Quantum Information Processing 39 5.1 Cubic gate - theoretical concept . 41 5.2 Experimental cubic states . 45 5.3 Streamlining the scheme . 48 5.4 Summary . 52 6 Conclusion 53 References 55 Supplementary material 65 Chapter 1 Introduction By the end of the 19th century, the common sentiment was that nearly all the mysteries of physics were already unraveled. True, there were few in- consistencies in need of ironing out, but after that was done there would be nothing left to do but perform increasingly precise measurements of funda- mental constants. Or so was thought. One of these minor quirks was the spectrum of blackbody radiation and its continuous and stubborn refusal to fully match the existing theoretical models. These differences were finally reconciled by introduction of photons [1] which ultimately lead to a com- pletely new field, a completely new physics - quantum physics. Originally, the photons were an abstract concept used to model statistical properties of light. Today, more than a century later, we are in an era in which in- dividual photons and their nonlocal properties can be observed and even manipulated with aims of accomplishing a number of intriguing goals. The research field with the ambition to harness the spooky nature of quantum systems in order to devise new and powerful technologies is called quantum information processing (QIP) and it is an amalgam of physics and information theory. So far, the investigation revealed the exciting pos- sibilities of quantum simulation [2, 3], quantum computation [4, 5, 6, 7], quantum communication [8, 9, 10], and quantum metrology [11, 12], but other paradigm shifting discoveries may be revealed in the future. All of the aforementioned applications require, as one of the elementary build- 1 ing blocks, quantum nonlinearity which is incompatible with classical and semi-classical descriptions. The nonlinear behavior can be found in natural systems, but it is unfortunately fairly weak and vulnerable to decoherence. Reliably providing desired nonlinear behavior is therefore one of the bottle- necks holding back the development of quantum technologies. Quantum technologies need to be firmly rooted in real physical systems and many experimental platforms are investigated for this purpose. Pho- tons or, more formally, traveling light are one of the analyzed possibilities. Light is a natural candidate for quantum connection [13] and the associ- ated information processing. Quantum states of light, either of individual photons or more complex multiphoton fields, can be straightforwardly gen- erated already with the present day technology. These states can be also effectively manipulated by linear optical schemes and linear squeezing, and measured by homodyne or single photon detectors [14, 15, 16]. Detectors resolving exact numbers of photons are not yet easily accessible but they are a focus of intensive investigation [17, 18]. One of light's main advantages is its resilience to noise. This is related to the ease with which a mode of light can be prepared in what is effectively the zero temperature state and isolated from a noisy environment during propagation. However, the nonin- teractibility of light, responsible for this robustness, is also the reason why realizing nonlinear operations for light is a fairly challenging task. Currently, the most feasible way of obtaining a nonlinear behavior for modes of traveling light lies in inducing it through a suitable measurement. Detectors of discrete quantum qualities, such as numbers of photons, are inherently nonlinear and, when used on a part of an entangled multi-mode field of light, are capable of probabilistically projecting the remaining state into a nonlinear one [19]. This is the traditional way of generating individual photons, but it was also used for creating of more complicated nonlinear states [20] and implementing quantum nonlinear operations [21]. This thesis expands upon this basic principle and shows several different kinds of nonlinearity which can be constructed at the single photon level. The theoretical proposals of the protocols form the the sturdy basis of the thesis. However, from the very first phase of their conception, they were 2 CHAPTER 1. INTRODUCTION always motivated by the experimental reality and the tools available. That is why a significant portion of the mainly theoretical thesis is composed of reports of the experiments which took the proposals and turned them into real life devices. The main results of the thesis can be separated into three thematically distinct topics, each one summarized in a separate chapter. In chapter 3 we will focus on a specific example of quantum nonlinear- ity - the noiseless amplification. We will introduce the problem and present specific proposals for implementation [A1]. We then demonstrate their feasi- bility by showcasing the experimental results obtained in collaboration with Max Planck Institute for Science of Light in Erlangen [A2, A3]. We will part with the chapter by abstracting the noiseless amplification and pondering its consequences for probabilistic measurements of quantum phase [A4]. In chapter 4 we will abandon the specific example and consider the broader scope of quantum information processing. We will show several ways in which we can manipulate a quantum state of light in an arbitrary way. Specifically, we will show a feasible set of elementary logic gates re- quired for processing of superposed coherent state qubits [A5] and discuss the experimental results obtained in collaboration with Danish Technical Institute in Lyngby [A6]. We then present a set of elementary gates suitable for a general transformation of optical field [A8] and discuss the nature of universal quantum resources required by these operations [A7]. As in chap- ter 3, all the operations considered in this chapter are probabilistic. This makes them unsuitable for universal computation and other protocols aim- ing for speed-up, but they are well fit for preparation of quantum states and tests of other tools in the quantum optical toolbox [16, 36]. In chapter 5 we will show how to implement a nonlinear quantum gate deterministically. Specifically, we will propose a realization of the cubic gate [A9, A12], which is the lowest order nonlinear gate sufficient for universal processing of quantum information. The key element required by the gate is a nonlinear quantum state used as a source of the nonlinearity. We will show that the required state can be constructed, photon by photon, with help of suitable measurements. We will analyze the experimental results obtained in collaboration with the University of Tokyo [A10, A11] and show that the 3 required nonlinearity is present in the generated states despite the existing experimental imperfections. 4 Chapter 2 Nuts and bolts of quantum optics Quantum information processing with quantum optics can be approached from two different angles. In the first one, the relevant physical systems are individual photons, which can be found in distinctive modes and labeled accordingly. The total number of these modes1, together with the number of photons, then determine the dimension of the Hilbert space used for de- scribing the system. This dimension can be arbitrarily large, but it is always finite and the total number photons of is always limited and well known. This is the historically older Discrete Variable (DV) approach [23], which has been already used for realization of many fundamental tests of quantum mechanics [24, 25, 26, 27], as well as for proof-of-principle test of quantum in- formation protocols [6, 28] and even first commercially available devices [30]. The strength of the DV approach lies in the natural self-correcting ability of the systems - a random loss of a photon can be easily detected and therefore does not introduce errors. However, it is not without weaknesses. There are experimental limits as to how many photons can be utilized at the same time and what operations can be realized by the available detectors. For example, quantum teleportation [31], the quintessential quantum protocol which plays role in many advanced applications, can not be deterministically 1polarization, spatial, temporal, and so on 5 realized without nonlinear operations which are currently unavailable in the DV quantum optics [32]. While the DV theory is governed by the algebra of finite Hilbert spaces, the DV experiments are fully determined by the available tools: sources of single photons, single photon detectors, and quantum operations preserv- ing photon numbers.
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