Spatial Characterization Of Two-Photon States - GAP-Optique

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1. General description of two-photon states In order to obtain more convenient units, the filter in momentum is defined in a different way than the filter in frequency. While Bs,i → 0 implies a single frequency collection, the condition for a single q vector collection is ws,i → ∞. Finally, after all the approximations and taking into account the filters, the mode function equation 1.12 becomes Φ(qs, Ωs, qi, Ωi) ∝ exp − w2 p 4 ∆20 − w2 p 4 ∆21 × exp − (γL)2 4 ∆2k − T 2 0 4 (Ωs + Ωi) 2 × exp − w2 s 2 q2 s − w2 i 2 q2 i − 1 2B2 Ω s 2 s − 1 2B2 Ω i 2 i . (1.34) The approximations listed in this chapter were introduced by several authors, and can be found for example in references [39, 40, 30]. In reference [41], we introduced a novel matrix notation based on the simplified mode function equation 1.34. The next section explains the details and the advantages of that notation. 1.4 The mode function in matrix form The argument of the exponential function in equation 1.34 is a second order polynomial of the mode function variables. Such a function can be written in matrix form, as is shown in appendix A. The mode function then becomes Φ(qs, Ωs, qi, Ωi) ∝ exp − 1 2 xt Ax (1.35) where all the parameters of the spdc process are contained in A, a positivedefinite real 6 × 6 matrix given by A = 1 ⎛ a ⎜ h ⎜ i 2 ⎜ j ⎝ k h b m n p i m c s t j n s d v k p t v f l r u w z ⎞ ⎟ , ⎟ ⎠ (1.36) l r u w z g x is the vector including all variables of the mode function, defined as ⎛ ⎞ ⎜ x = ⎜ ⎝ q x s q y s q x i q y i Ωs Ωi ⎟ , (1.37) ⎟ ⎠ and x t is the transpose of x. This matrix representation of the mode function, introduced in reference [41], is an important result of this thesis. The use of this notation reduces the calculation time for integrals over the mode function, 12
1.4. The mode function in matrix form and allows to solve some of those integrals analytically. For instance, the mode function normalization requires six integrals over the modulus square of the mode function. As appendix B shows, the matrix notation provides a way to calculate the normalization constant analytically, dx exp − 1 2 xt (2A)x and thus the normalized mode function reads = Φ(qs, Ωs, qi, Ωi) = [det(2A)]1/4 (2π) 3/2 (2π) 3 , (1.38) [det(2A)] 1/2 exp − 1 2 xt Ax . (1.39) In the same way, many other integrals over the mode function can be solved analytically using the matrix representation of the mode function. Conclusion The two-photon mode function can be written in a matrix form after a series of approximations. The matrix notation makes it possible to calculate several features of the down-conversion photons analytically, and reduces the numerical calculation time of others. The next chapter describes the different correlations in the two-photon state using the purity as a correlation indicator. The use of the approximate mode function in matrix notation makes it possible to find an analytical expression for the purity, and to study the effect of the different parameters on it. 13
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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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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 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: 1 0 -0.2 1.3. Approximations and ot
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
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
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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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