Spatial Characterization Of Two-Photon States - GAP-Optique

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When he [Kepler] found that his long cherished beliefs did not agree with the most precise observations, he accepted the uncomfortable facts, he preferred the hard truth to his dearest illusions. That is the heart of science. Cosmos - Carl Sagan
Abstract In the same way that electronics is based on measuring and controlling the state of electrons, the technological applications of quantum optics will be based on our ability to generate and characterize photonic states. The generation of photonic states is traditionally associated to nonlinear optics, where the interaction of a beam and a nonlinear material results in the generation of multi-photon states. The most common process is spontaneous parametric down-conversion (spdc), which is used as a source of pairs of photons not only for quantum optics applications but also for quantum information and quantum cryptography [1, 2]. The popularity of spdc lies in the relative simplicity of its experimental realization, and in the variety of quantum features that down-converted photons exhibit. For instance, a pair of photons generated via spdc can be entangled in polarization [3, 4], frequency [5, 6], or in the equivalent degrees of freedom of orbital angular momentum, space, and transverse momentum [7, 8, 9, 10, 11]. Standard spdc applications focus on a single degree of freedom, wasting the entanglement in other degrees of freedom and the correlations between them. Among the few configurations using more than one degree of freedom are hyperentanglement [12, 13], spatial entanglement distillation using polarization [14], or control of the joined spectrum using the pump’s spatial properties [6]. This thesis describes the spatial properties of the two-photon state generated via spdc, considering the different parameters of the process, and the correlations between space and frequency. To achieve this goal, I use the purity to quantify the correlations between the photons, and between the degrees of freedom. Additionally, I study the spatial correlations by describing the transfer of orbital angular momentum (oam) from the pump to the signal and idler photons, taking into account the pump, the detection system and other parameters of the process. This thesis is composed of six chapters. Chapter 1 introduces the mode function, used throughout the thesis to describe the two-photon state in space and frequency. Chapter 2 describes the correlations between degrees of freedom or photons in the two-photon state, using the purity to quantify such correlations. Chapter 3 explains the mechanism of the oam transfer from pump to signal and idler photons. Chapter 4 describes the effect of different spdc parameters on the oam transfer in noncollinear configurations, both theoretically and experimentally. In analogy with the downconverted case, chapter 5 discusses the two-photon state generated via Raman transition, by describing its mode function, the correlations between different parts of the state, and the oam transfer in the process. Finally, chapter 6 summarizes the main results presented by this thesis. xi
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DOCTORAL THESIS IN PHOTONICS ICFO B
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Spatial Characterization Of Two-Pho
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Contents Contents vii Acknowledgeme
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Acknowledgements This thesis compil
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Abstract In the same way that elect
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Resumen De la misma manera que la e
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Introduction The role of photons in
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This thesis is based on the followi
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1. General description of two-photo
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CHAPTER 2 Correlations and entangle
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2.1. The purity as a correlation in
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2.2. Correlations between space and
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2.2. Correlations between space and
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2.3. Correlations between signal an
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2.3. Correlations between signal an
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Signal purity 1 0 (a) 0.1 3 2.3. Co
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CHAPTER 3 Spatial correlations and
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3.2. OAM transfer in general SPDC c
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3.2. OAM transfer in general SPDC c
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Phase front 3.3. OAM transfer in co
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4. OAM transfer in noncollinear con
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4. OAM transfer in noncollinear con
Page 62 and 63: 4. OAM transfer in noncollinear con
Page 72 and 73: 5. Spatial correlations in Raman tr
Page 81 and 82: CHAPTER 6 Summary This thesis chara
Page 83 and 84: APPENDIX A The matrix form of the m
Page 85 and 86: v =γ 2 L 2 Np sin ϕi − γ 2 L 2
Page 87: The matrix C is given by C = 1 ⎛
Page 90 and 91: B. Integrals of the matrix mode fun
Page 92 and 93: C. Methods for OAM measurements Inp
Page 94 and 95: Bibliography [13] M. Barbieri, C. C
Page 96 and 97: Bibliography [41] C. I. Osorio, A.
Page 98: Bibliography [68] S. Chen, Y.-A. Ch
Page 103: Spatial Characterization Of Two-Pho
Page 107: A Luz Stella y Luis Alfonso, mis pa
Page 110 and 111: Contents B Integrals of the matrix
Page 114 and 115: Abstract The matrix notation, intro
Page 116 and 117: Abstract La notación matricial int
Page 118 and 119: Introduction the transfer of oam fr
Page 121 and 122: CHAPTER 1 General description of Tw
Page 123 and 124: y y x z z pump s i (a) signal idle
Page 125 and 126: 1.3. Approximations and other consi
Page 127 and 128: x 1.3. Approximations and other con
Page 129 and 130: x Wave fronts 1.3. Approximations a
Page 131 and 132: 1 0 -0.2 1.3. Approximations and ot
Page 133: 1.4. The mode function in matrix fo
Page 136 and 137: 2. Correlations and entanglement (a
Page 138 and 139: 2. Correlations and entanglement ch
Page 140 and 141: 2. Correlations and entanglement th
Page 142 and 143: 2. Correlations and entanglement Fi
Page 144 and 145: 2. Correlations and entanglement q
Page 146 and 147: 2. Correlations and entanglement Si
Page 148 and 149: 2. Correlations and entanglement ri
Page 150 and 151: 3. Spatial correlations and OAM tra
Page 157 and 158: CHAPTER 4 OAM transfer in noncollin
Page 159 and 160: 4.2. Effect of the pump beam waist
Page 161 and 162: 1.0 0.0 4.2. Effect of the pump bea
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Coincidences 800 0 -4 4.2. Effect o
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Weight 1 0 4.3. Effect of the Poynt
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4.3. Effect of the Poynting vector
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Before the crystal 4.3. Effect of t
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CHAPTER 5 Spatial correlations in R
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x y z 5.1. The quantum state of Sto
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where 5.2. Orbital angular momentum
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weight 1 0.6 -180º -90º 0º 90º
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5.3. Spatial entanglement 5.20 is s
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6. Summary Experiments that explain
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A. The matrix form of the mode func
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A. The matrix form of the mode func
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APPENDIX B Integrals of the matrix
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APPENDIX C Methods for OAM measurem
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Bibliography [1] M. A. Nielsen and
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Bibliography [27] J. P. Torres, A.
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Bibliography [55] M. C. Booth, M. A
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