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Verification of Nonclassicality in Phase Space Verification of nonclassicality in phase space Dissertation zur Erlangung des akademischen Grades doctor rerum naturalium (Dr. rer. nat.) der Mathematisch-Naturwissenschaftlichen Fakult¨at der Universit¨atRostock vorgelegt von Thomas Kiesel, geb. am 07.02.1983 in Wismar aus Rostock Rostock, 29. Juni 2011 Gutachter: Prof. Dr. Werner Vogel, Universit¨atRostock Prof. Dr. John Klauder, University of Florida Prof. Dr. Adam Miranowicz, University Pozna´n Tag der Verteidigung: 12. Oktober 2011 Abstract Nonclassicality has been introduced to examine the question whether experimental observations can be explained within a classical physical theory. In quantum op- tics, it is defined as the failure of the so-called Glauber-Sudarshan quasiprobability to be interpreted as a classical probability density. Although this definition has been accepted for several decades, there has been a lack of simple and complete criteria to check whether a given quantum state of light is nonclassical. In this thesis, the experimental use of some nonclassicality criteria is examined. Then, so-called nonclassicality quasiprobabilities are introduced, which allow a complete investigation of nonclassicality of an arbitrary quantum state in a simple manner. Their theoretical background is carefully elaborated, and their use is demonstrated with experimental data. Finally, some issues of the relation between nonclassicality and the special effect of entanglement are discussed, again with the application of experimental measurements. Zusammenfassung Der Begriff der Nichtklassizit¨atwurde eingef¨uhrt, um Antworten auf die Frage zu finden, ob bestimmte experimentelle Ergebnisse noch mit einer klassischen physi- kalischen Theorie erkl¨artwerden k¨onnen.In der Quantenoptik ist Nichtklassizit¨at dadurch definiert, dass die sogenannte Glauber-Sudarshan Quasiverteilung nicht mehr die Eigenschaften einer klassischen Wahrscheinlichkeitsverteilung erf¨ullt.Ob- wohl dieses Ph¨anomenbereits seit einigen Jahrzehnten diskutiert wird, fehlten im- mer noch einfache Kriterien, mit denen man die Frage der Nichtklassizit¨atf¨ur einen gegebenen Quantenzustand des Lichts vollst¨andigbeantworten kann. In der vorliegenden Dissertation werden einige Kriterien f¨urNichtklassizit¨at auf ihre expe- rimentelle Anwendbarkeit untersucht. Dann werden sogenannte Nichtklassizit¨ats- Quasiverteilungen eingef¨uhrt,die eine vollst¨andigeUntersuchung der Nichtklas- sizit¨ateines beliebigen Quantenzustands mit einfachen Mitteln erlauben. Ihre the- oretischen Grundlagen werden sorgf¨altig ausgearbeitet, und die Praktikabilit¨atder Methode wird mit experimentell vermessenen Zust¨andenunter Beweis gestellt. Am Ende werden einige Zusammenh¨angezwischen Nichtklassit¨atund ihrem weit ver- breiteten Spezialfall, der Verschr¨ankung,diskutiert, wobei ebenfalls experimentelle Daten geeigneter Zust¨andeverwendet werden. Contents I. Dissertation thesis 1 1. Introduction 3 2. Phase space methods in quantum mechanics 5 2.1. Coherent states . .5 2.2. Weyl symbols of operators . .6 2.3. Glauber-Sudarshan- and generalized quasiprobabilities . .7 2.4. Experimental reconstruction of quasiprobabilities . .9 3. Nonclassicality of the harmonic oscillator 13 3.1. Definition of nonclassicality . 13 3.2. Nonclassicality criteria . 14 3.2.1. Nonclassicality witnesses . 14 3.2.2. Matrices of moments and characteristic functions . 16 3.2.3. Comparison of different nonclassicality criteria . 18 3.2.4. Negativities of quasiprobabilities . 19 4. Nonclassicality quasiprobabilities 25 4.1. Necessity of complete nonclassicality criteria . 25 4.1.1. Nonclassicality criteria for pure states . 25 4.1.2. Nonclassicality criteria for mixed states . 27 4.2. Nonclassicality filters and quasiprobabilities . 27 4.3. Construction of nonclassicality filters . 30 4.4. Relation to different filtering procedures . 31 4.5. Application to a single-photon added thermal state . 32 4.6. Direct sampling of nonclassicality quasiprobabilities . 33 5. Experimental bipartite entanglement verification 37 5.1. Entanglement and its relation to nonclassicality . 37 5.1.1. Definition of bipartite entanglement . 37 5.1.2. On quasiprobabilities for the detection of entanglement . 38 5.1.3. Entanglement criteria . 39 Contents 8 5.2. Nonclassicality and entanglement at a beam splitter . 40 6. Outlook 45 II. Publications 63 Part I. Dissertation thesis 1. Introduction What is the difference between classical physics and the quantum world? Is it necessary to apply quantum mechanics to describe the behavior of matter and light? Do we have to discuss fields of light, which are not in a specific state, but in a superposition of many different states? Or is it possible to develop a theory, founded on classical electrodynamics and supplemented with a statistical framework, which also describes all experimental phenomena? These questions have been of great interest from the early rise of quantum theory, since they are related to our deep understanding of physics. For instance, we may hardly imagine Schr¨odingerscat, which is supposed to be in a superposition of two states, \dead" or \alive", but does it already mean that it is impossible to describe this situation in terms of classical physics? In quantum optics, these issues are closely related to the notion of nonclassica- lity. Expressing it in a simple way, a nonclassical state is a state of the optical field of light, whose properties cannot be fully described in terms of classical electrody- namics. More precisely, these properties are correlation functions of the electro- magnetic field, which can be measured with specific optical devices. If one is not able to derive these outcomes of the measurement from classical theory, the state must have some typically quantum properties. Hence, quantum optical methods are necessary for its characterization. The term nonclassicality has been defined decades ago by Titulaer and Glauber [1] as the failure of the so-called Glauber-Sudarshan P function [2, 3] to exhibit the properties of a classical probability density. They have seen that the P function can be formally used as probability density to calculate expectation values of quantum optical observables, but may have negativities, which is impossible for a probabi- lity density. Furthermore, they showed that nonclassical states possess certain field correlation functions, which can be measured in experiments, but not explained by classical physics. Therefore, nonclassicality is a signature of quantumness. Although this definition is known for several decades, its practical application is difficult, since the Glauber-Sudarshan P function is highly singular for many states, which prevents it from experimental reconstruction. To overcome this prob- lem, many different signatures of nonclassicality have been derived, such as squee- zing [4] or antibunching [5]. However, they only provide sufficient criteria for a state to be nonclassical. On the contrary, there are also sets of complete criteria of nonclassicality, which are based on determinants of moments [6] or characteristic functions of the state [7]; but these criteria are based on infinite hierarchies of in- 1 Introduction 4 equalities, which cannot be checked all in practice. Therefore, there existed a lack of a complete, but still simple criterion for the nonclassicality of a quantum state. The main goal of this thesis is to tackle this problem. Our solution is the intro- duction of so-called nonclassicality probabilities, which overcome the singularities of the P function, but indicate nonclassical effects by negativities. We show that it is sufficient to equip the nonclassicality quasiprobability with a single real width parameter in order to derive a complete criterion for the verification of nonclassica- lity. In this sense, our criterion is similar to the original definition. Moreover, the nonclassicality quasiprobabilities are designed to be accessible from experimental data, and their application is demonstrated in this work. For this purpose, we begin with an introduction to phase-space methods in quan- tum optics. These are necessary for the definition of nonclassicality. After a review of some of the previously known nonclassicality criteria and an examination of their experimental application, we proceed with the reconstruction of the P function of a nonclassical state, which is approximately possible for certain states. Then we in- troduce the concept of nonclassicality quasiprobabilities, which overcomes problems of the previous approaches, and demonstrate their experimental applicability. At the end of this thesis, we make a small excursion to the verification of en- tanglement. This is a special nonclassical phenomenon of particular interest, but has to be treated in a completely different manner. This work provides some ad- ditional investigation of the features of nonclassicality and the relation between nonclassicality and entanglement. 2. Phase space methods in quantum mechanics We start our considerations with a general introduction into the phase-space for- malism of quantum mechanics. We begin with a short description of coherent states, which provide the basis for phase-space functions. Then we explain the definition and use of symbols, which are complex-valued functions representing the quantum mechanical operators. The Glauber-Sudarshan P function, being the key for the definition of nonclassicality, is a special symbol of the density operator of a quantum state. Finally, we conclude with some remarks on the experimental re- construction of phase-space functions of the density
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