Spatial Characterization Of Two-Photon States - GAP-Optique

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2. Correlations and entanglement q s x q s y q i x y qi s i qs x a h i j k l q s y h b m n p r qi y i m c s t u Figure 2.5: The signal-idler cross terms in matrix A are responsible for the correlations between the generated photons. By suppressing these cross terms, and by making permutations over columns and rows, the matrix becomes a two block matrix, where each block contains the information of one photon. ing the following terms negligible i =w 2 p + γ 2 L 2 tan ρ0 2 j =γ 2 L 2 sin ϕi tan ρ0 l =γ 2 L 2 tan ρ0(Np − Ni cos ϕi) m = − γ 2 L 2 sin ϕs tan ρ0 q i x n = − γ 2 L 2 sin ϕs sin ϕi + w 2 p cos ϕs cos ϕi r = − γ 2 L 2 sin ϕs(Np − Ni cos ϕi) + w 2 pNi cos ϕs sin ϕi t =γ 2 L 2 tan ρ0(Np − Ns cos ϕs) v =γ 2 L 2 sin ϕi(Np − Ns cos ϕs) − w 2 pNs cos ϕi sin ϕs z =γ 2 L 2 N 2 p − γ 2 L 2 Np(Ni cos ϕi + Ns cos ϕs) + γ 2 L 2 NsNi cos ϕs cos ϕi j n s d v w + T 2 0 − w 2 pNsNi sin ϕs sin ϕi. (2.15) As an example, a configuration with a negligible Poynting walk-off fulfills this condition always when cos ϕs = Np 2Ns cos ϕi = Np 2Ni w 2 p γ 2 L 2 = tan ϕs tan ϕi T 2 0 =N 2 p γ 2 L 2 wp ≪ws, wi s k p t v f z i l r u w tan ϕs 2 tan ϕi 2 − 1 4 z g (2.16) When the correlations are totally suppressed by satisfying these or other conditions, the two-photon state is separable, and each photon is in a pure state. Therefore, the two-photon mode function can be written as a product of two mode functions, one for each photon, 24 Φ(q s, ωs, q i, ωi) = Φs(q s, Ωs)Φi(q i, Ωi). (2.17)
2.3. Correlations between signal and idler In any other case, correlations between signal and idler are unavoidable. To evaluate their strength, the next section deduces an analytical expression for the purity of the signal photon state. 2.3.2 Spatio temporal state of signal photon A single photon state can be generated in spdc by ignoring any information about one of the generated photons. The single photon state is calculated by tracing out the other photon from the two-photon state. For example, after a partial trace over the idler photon, the reduced density matrix in space and frequency for the signal photon is ˆρsignal =T ridler[ρ] = = and its purity is given by dq ′′ i dΩ ′′ i 〈q ′′ i , Ω ′′ i |ˆρ|q ′′ i , Ω ′′ i 〉 dqsdΩsdqidΩidq ′ sdΩ ′ s Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)|qs, Ωs〉〈q ′ s, Ω ′ s|, (2.18) T r[ˆρ 2 signal] = dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i × Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ω ′ s, qi, Ωi) × Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i). (2.19) Recalling the exponential character of the mode function described by equation 1.35, equation 2.19 writes T r[ˆρ 2 signal] = det(2A) det(C) , (2.20) where C is a positive-definite real 12 × 12 matrix defined by N 4 exp − 1 2 Xt CX = Φ(qs, Ωs, qi, Ωi) and given by ⎛ C = 1 2 ⎜ ⎝ × Φ ∗ (q ′ s, Ω ′ s, qi, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ωs, q ′ i, Ω ′ i), (2.21) 2a 2h i j 2k l 0 0 i j 0 l 2h 2b m n 2p r 0 0 m n 0 r i m 2c 2s t 2u i m 0 0 t 0 j n 2s 2d v 2w j n 0 0 v 0 2k 2p t v 2f z 0 0 t v 0 z l r 2u 2w z 2g l r 0 0 z 0 0 0 i j 0 l 2a 2h i j 2k l 0 0 m n 0 r 2h 2b m n 2p r i m 0 0 t 0 i m 2c 2s t 2u j n 0 0 v 0 j n 2s 2d v 2w 0 0 t v 0 z 2k 2p t v 2f z l r 0 0 z 0 l r 2u 2w z 2g ⎞ ⎟ . ⎟ ⎠ (2.22) 25
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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
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 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 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
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
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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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