Spatial Characterization Of Two-Photon States - GAP-Optique

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2. Correlations and entanglement Filters 0nm 1nm 10nm →∞ 0.1 Collection mode 1mm <strong>Spatial</strong> purity Figure 2.3: The spatial purity increases by filtering the state in space or frequency. For the parameters chosen here, the bandwidth of the generated photons is smaller than 10 nm, making 10 nm filters equivalent to infinitely broad ones. The parameters used to generate this figure are listed in table 2.1. <strong>Spatial</strong> purity 1 0 0.1 Pump waist 30m 462m Emission angle 1mm 0.1 Collection mode 1mm 1 0.2 1º 5º (a) (b) (c) 0º 0.1 1mm Pump waist 20m 500m Figure 2.4: For the values reported in reference [6], figure (a) shows that the purity is smaller than 1. This fact is in agreement with the implicit spatio-temporal correlations reported in the reference. For the values reported in references [46] and [31], figures (b) and (c) show that the purity has its maximum value. There was no evidence of spatio-temporal correlations in the results reported in these references. In all three cases, it is possible to modify the value of the purity by tailoring the spdc parameters. The parameters used to generate this figure are listed in table 2.1. 10 nm interference filters. For the noncollinear configuration in reference [31], the highly focused pump beam is responsible for the lack of correlations. The figure shows that for lessfocused pump beams the purity decreases as the correlations between space and frequency become more important. The correlations between frequency and space in the two-photon state discussed in this section, suggest that a complete description of the correlations between the generated photons, should take into account both degrees of freedom, as the next section discusses. 22
2.3. Correlations between signal and idler Table 2.1: The parameters used in figures 2.3 and 2.4 Parameter Figure 2.3 Figure 2.4(a) Figure 2.4(b) Figure 2.4(c) Crystal liio3 liio3 bbo bbo L 1 mm 1 mm 1.5 mm 2 mm ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦ T0 → ∞ → ∞ → ∞ → ∞ wp 400 µm 30 / 462 µm wp ≫ L 20, 500 µm λp 405 nm 405 nm 405 nm 351.1 nm ∆λp 0 0.4 nm λ 0 s 810 nm 810 nm 810 nm 702 nm ∆λs 0.2 nm 10 nm 10 nm ϕs 10 ◦ 17 ◦ 0, 1, 5 ◦ 4 ◦ 2.3 Correlations between signal and idler In contrast to the previous section, here the two-photon system is considered as composed by two photons, each described by space and time variables, as figure 2.1 (b) shows. Since the global state in equation 1.9 is pure, the purity of the signal photon calculated in this section is a measurement of the degree of spatiotemporal entanglement between signal and idler photons, as in references [5, 10, 11, 47, 48]. spdc as a source of two-photon states can be used to generate pairs of photons maximally entangled or two single photons [5, 49]. The purity of the signal photon in the first case should be equal to 0 and in the second case equal to 1. This section explores the necessary conditions to be in each regime. This section is divided in three parts, the first one discusses the origin of the correlations and the strategies to suppress them. The second part describes how to obtain an analytical expression for the signal photon purity. And the third part shows the results of numerical calculation of that purity. 2.3.1 Origin of the correlations between photons The presence of cross terms in variables of both signal and idler in equation 1.34 indicates correlations between the generated photons. Figure 2.5 shows those terms: i, j, l, m, n, r, t, v, and z. In contrast to the last section, filtering one degree of freedom is not enough to suppress the correlations between signal and idler. For example, by using a frequency filter, the photons can be still correlated in space. The total suppression of the correlation requires infinite narrow filters in both degrees of freedom for one of the photons, as shown experimentally in reference [50]. But that kind of filtering reduces the number of available pairs of photons substantially. The other possibility to generate uncorrelated photons is to tailor the parameters of the spdc process to make the cross terms negligible simultaneously. According with appendix A, when α = 0 ◦ , this condition is equivalent to mak- 23
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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
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 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 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
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
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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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