Spatial Characterization Of Two-Photon States - GAP-Optique

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5. <strong>Spatial</strong> correlations in Raman transitions Schmidt number 40 20 1 -180º 0º 180º lenght spatial filter 2 mm 1 mm 400 m 200 m 7 4 1 -180º 0º 180º 100 m 200 m 500 m angle Figure 5.5: The length of the cloud controls the Schmidt number angular dependence. By modifying this length it is possible to change the position of the maximum and minimum values of the Schmidt number. The angular dependence can be removed completely by tailoring the parameters of the process, or by filtering. Table 5.1 lists the parameters used to generate this figure. Table 5.1: The parameters used in figures 5.3, 5.4 and 5.5 Parameter Figure 5.3 Figure 5.4 Figure 5.5 (left) Figure 5.5 (right) L 0.2, 0.4, 1, 2 mm [0.2, 1.5] mm 0.2, 0.4, 1, 2 mm 200 µm wp 100 µm 100 µm 500 µm 500 µm R 400µm 400µm 1000 µm 1000 µm ws 100 µm 100 µm 100 µm 100, 200, 500 µm wg 500 µm 500 µm ϕs [−180, 180] ◦ 0, 10, 90 ◦ [−180, 180] ◦ [−180, 180] ◦ section shows the effects of changing the emission angle on the Schmidt number K defined in section 2.1, is a common entanglement quantifier. The Schmidt number of the two-photon state described by equation 5.12 quantifies the entanglement between the generated photons. According to references [69, 70] it is given by (A + B) (C + D) K = 4 (ABCD) 1/2 . (5.21) Figure 5.5 shows the Schmidt number as a function of the emission angle for different values of the atomic cloud length and the pump beam waist. The function extrema are located at 0 ◦ , 90 ◦ , 180 ◦ and 270 ◦ . At each of these values the function can have a maximum or a minimum depending on the other parameters of process. For instance, if L < R 1 + 2R2 w2 −1/2 p (5.22) the entanglement is maximum at collinear configurations where ϕ = 0 ◦ , 180 ◦ , and minimum at transverse configurations ϕ = 90 ◦ , 270 ◦ . In the left part of figure 5.5, the condition in equation 5.22 is achieved at cloud lengths smaller than 333 µm. As the length of the cloud increases, the variation of the entanglement with the angle is smoothed out. In the left part of figure 5.5 the relation in equation 58
5.3. <strong>Spatial</strong> entanglement 5.20 is satisfied when L = 333 µm; therefore, the amount of entanglement is constant. If the length of the cloud increases even more, the position for the maxima and the minima get inverted. When L > R 1 + 2R2 w2 −1/2 p (5.23) the entanglement is maximum for transverse emitting configurations ϕ = 90 ◦ , 270 ◦ , and minimum for collinear configurations ϕ = 0 ◦ , 180 ◦ . This variation is possible in Raman transitions where the transversal size of the cloud is comparable to the longitudinal size. Another parameter that plays a role in the variation of the amount of entanglement is the Stokes spatial filter ws. The right side of figure 5.5 shows the effect of changing the filtering over the amount of entanglement. Very narrow spatial filters (ws → ∞) diminish both the amount of entanglement and its azimuthal variability. Conclusion As in spdc, the geometrical configuration of the Raman transitions determines the oam content and the spatial correlations of the generated Stokes and anti- Stokes photons. The size and shape of the cloud defines the emission angles for which the correlations are maximum and minimum. 59
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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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Page 26 and 27:
Page 28 and 29: 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 and 44: 2.3. Correlations between signal an
Page 45 and 46: 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
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
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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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