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
Page 1: DOCTORAL THESIS IN PHOTONICS ICFO B
Page 5: Spatial Characterization Of Two-Pho
Page 9 and 10: Contents Contents vii Acknowledgeme
Page 11 and 12: Acknowledgements This thesis compil
Page 13 and 14: Abstract In the same way that elect
Page 15 and 16: Resumen De la misma manera que la e
Page 17 and 18: Introduction The role of photons in
Page 19: This thesis is based on the followi
Page 22 and 23: 1. General description of two-photo
Page 35 and 36: CHAPTER 2 Correlations and entangle
Page 37 and 38: 2.1. The purity as a correlation in
Page 39 and 40: 2.2. Correlations between space and
Page 41 and 42: 2.2. Correlations between space and
Page 43: 2.3. Correlations between signal an
Page 47 and 48: Signal purity 1 0 (a) 0.1 3 2.3. Co
Page 49 and 50: CHAPTER 3 Spatial correlations and
Page 51 and 52: 3.2. OAM transfer in general SPDC c
Page 53 and 54: 3.2. OAM transfer in general SPDC c
Page 55: Phase front 3.3. OAM transfer in co
Page 58 and 59: 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
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Bibliography [41] C. I. Osorio, A.
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Bibliography [68] S. Chen, Y.-A. Ch
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Spatial Characterization Of Two-Pho
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A Luz Stella y Luis Alfonso, mis pa
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Contents B Integrals of the matrix
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When he [Kepler] found that his lon
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Abstract The matrix notation, intro
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Abstract La notación matricial int
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Introduction the transfer of oam fr
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CHAPTER 1 General description of Tw
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y y x z z pump s i (a) signal idle
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1.3. Approximations and other consi
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x 1.3. Approximations and other con
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x Wave fronts 1.3. Approximations a
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1 0 -0.2 1.3. Approximations and ot
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1.4. The mode function in matrix fo
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2. Correlations and entanglement (a
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2. Correlations and entanglement ch
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2. Correlations and entanglement th
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2. Correlations and entanglement Fi
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2. Correlations and entanglement q
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2. Correlations and entanglement Si
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2. Correlations and entanglement ri
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3. Spatial correlations and OAM tra
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CHAPTER 4 OAM transfer in noncollin
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4.2. Effect of the pump beam waist
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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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Spatial Characterization Of Two-Photon States - GAP-Optique

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