Spatial Characterization Of Two-Photon States - GAP-Optique

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2. Correlations and entanglement rity as correlation indicator it is possible to determine the conditions in which the specific correlations are suppressed, without requiring the use of infinitely narrow filters. The next chapter continues with the description of the correlations between photons, in the case where only their spatial state is relevant. The chapter describes the spatial correlations in terms of orbital angular momentum transfer. 28
CHAPTER 3 <strong>Spatial</strong> correlations and orbital angular momentum transfer The previous chapter describes the correlations between the signal and idler photons considering space and frequency, as well as the correlations between those degrees of freedom. This chapter focuses on the cases where the degrees of freedom are not correlated, and the only relevant correlations are those between the photons in the spatial degree of freedom. To characterize those correlations I use the orbital angular momentum (oam) content of the generated photons, which is directly associated to their spatial distribution. The spatial correlations between the photons determine the mechanism of oam transfer from the pump to the signal and idler. Most studies of the oam correlations in spdc can be divided in two categories: those reporting a full transference of the pump’s oam [7, 8, 22, 27, 28, 29], and those reporting a partial transference [30, 31, 32, 33]. This chapter describes the oam transfer mechanism, and shows the difference between both regimes. The chapter is divided in three sections. Section 3.1 introduces the eigenstates of the oam operator: the Laguerre-Gaussian modes. This section shows how to calculate and measure the oam content of the photons. Section 3.2 describes how the oam is transferred in spdc. As an example, section 3.3 shows how this transfer happens in the collinear case, where all the emitted photons propagate in the same direction. The main result of this chapter is a selection rule that summarizes the oam transfer mechanism: the oam carried by the pump is completely transferred to all the signal and idler photons emitted over the cone. 29
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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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5. Spatial correlations in Raman tr
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CHAPTER 6 Summary This thesis chara
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APPENDIX A The matrix form of the m
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v =γ 2 L 2 Np sin ϕi − γ 2 L 2
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The matrix C is given by C = 1 ⎛
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B. Integrals of the matrix mode fun
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C. Methods for OAM measurements Inp
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Bibliography [13] M. Barbieri, C. C
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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 112 and 113: When he [Kepler] found that his lon
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 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
Page 163 and 164: Coincidences 800 0 -4 4.2. Effect o
Page 165 and 166: Weight 1 0 4.3. Effect of the Poynt
Page 167 and 168: 4.3. Effect of the Poynting vector
Page 169 and 170: Before the crystal 4.3. Effect of t
Page 171 and 172: CHAPTER 5 Spatial correlations in R
Page 173 and 174: x y z 5.1. The quantum state of Sto
Page 175 and 176: where 5.2. Orbital angular momentum
Page 177 and 178: weight 1 0.6 -180º -90º 0º 90º
Page 179: 5.3. Spatial entanglement 5.20 is s
Page 182 and 183: 6. Summary Experiments that explain
Page 184 and 185: A. The matrix form of the mode func
Page 186 and 187: A. The matrix form of the mode func
Page 189 and 190: APPENDIX B Integrals of the matrix
Page 191 and 192: APPENDIX C Methods for OAM measurem
Page 193 and 194: Bibliography [1] M. A. Nielsen and
Page 195 and 196: Bibliography [27] J. P. Torres, A.
Page 197 and 198: Bibliography [55] M. C. Booth, M. A
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