Spatial Characterization Of Two-Photon States - GAP-Optique

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2. Correlations and entanglement the spatial two-photon state is ˆρq = T rΩ[ρ] = = dΩ ′′ s dΩ ′′ i 〈Ω ′′ s , Ω ′′ i |ˆρ|Ω ′′ s , Ω ′′ i 〉 dqsdΩsdqidΩidq ′ sdq ′ i Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)|qs, qi〉〈q ′ s, q ′ i|, (2.10) while the purity of this state, defined as T r[ˆρ 2 q], is given by T r[ˆρ 2 q] = dqsdΩsdqidΩidq ′ sdΩ ′ sdq ′ idΩ ′ i × Φ(qs, Ωs, qi, Ωi)Φ ∗ (q ′ s, Ωs, q ′ i, Ωi) × Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i). (2.11) It is possible to solve these integrals analytically by using the exponential character of the mode function given by equation (1.35). The purity becomes T r[ˆρ 2 q] = det(2A) det(B) . (2.12) As seen in appendix A, B is a positive-definite real 12 × 12 matrix B = 1 ⎛ ⎜ 2 ⎜ ⎝ 2a 2h 2i 2j k l 0 0 0 0 k 2h 2b 2m 2n p r 0 0 0 0 p 2i 2m 2c 2s t u 0 0 0 0 t 2j 2n 2s 2d v w 0 0 0 0 v k p t v 2f 2z k p t v 0 l r u w 2z 2g l r u w 0 0 0 0 0 k l 2a 2h 2i 2j k 0 0 0 0 p r 2h 2b 2m 2n p 0 0 0 0 t u 2i 2m 2c 2s t 0 0 0 0 v w 2j 2n 2s 2d v k p t v 0 0 k p t v 2f l r u w 0 0 l r u w 2z l r u w 0 0 l r u w 2z 2g defined by the equation N 4 exp − 1 2 Xt BX = Φ(qs, Ωs, qi, Ωi) ⎞ ⎟ , ⎟ ⎠ (2.13) × Φ ∗ (q ′ s, Ωs, q ′ i, Ωi)Φ(q ′ s, Ω ′ s, q ′ i, Ω ′ i)Φ ∗ (qs, Ω ′ s, qi, Ω ′ i), (2.14) where vector X is the result of concatenation of x and x ′ . In order to see the effect of each of the spdc parameters on the spatiofrequency correlations, some typical values are used in next subsection to calculate T r[ˆρ 2 q] numerically. 2.2.3 Numerical calculations As first example, consider a degenerate type-i spdc process characterized by the parameters in the second column of table 2.1. A pump beam, with wavelength 20
2.2. Correlations between space and frequency λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength λ0 s = λ0 i = 810 nm at an angle ϕs,i = 10◦ . Under the above conditions, figure 2.3 shows the purity of the spatial state T r[ρ2 q] given by equation 2.12, as a function of the spatial filter width for different frequency filter bandwidths. The half width at 1/e of the frequency filters ∆λn, is Bn = πc∆λn/(λ2 √ n ln 2). Figure 2.3 shows the presence of correlations between space and frequency in all the situations for which the purity is smaller than one. According to figure 2.3, the spatial purity of the two-photon state gets closer to one as ws (= wi) increase. Narrow filters in space (infinitely large ws,i) makes the two-photon state separable in frequency and space. The separability and the lack of correlations between frequency and space appear also in the“narrow band” limit for the frequency. If ∆λs = ∆λi → 0 nm, then the purity is always 1 for any value of ws = wi. When using typical interference filters, ∆λs = ∆λi ≈ 1 nm, the purity decreases indicating space/frequency correlations. As the width of the frequency filters increases, the purity converges quickly, the case of ∆λs = ∆λi = 10 nm is almost equivalent to the case without frequency filters ∆λs, ∆λi → ∞. In order to compare the theoretical results of this section with previous experimental work, figure 2.4 shows the purity of the spatial state as a function of the collection mode for three reported experiments. Figure 2.4 (a) shows the purity for the data reported in reference [6]. The authors of that paper studied a type-i spdc process in a 1 mm long liio3 crystal. They used a diode laser with wavelength λp = 405 nm and bandwidth ∆λp = 0.4 nm as a pump beam. And, by using monochromators with ∆λs,i = 0.2 nm, they detected pairs of photons emitted at ϕs,i = 17◦ . Table 2.1 resumes these parameters. The paper analyzes two pump waist values: wp = 30 µm and wp = 462 µm. For those values, the authors demonstrated control of the frequency correlations by changing the spatial properties of the pump beam. As the pump beam influences the spatial shape of the generated photons, they implicitly reported a correlation between space and frequency. This spatial to spectral mapping exists at their collection mode ws = 133.48 µm for which the correlation indeed appears according with our model. Figure 2.4(b) shows T r[ρ2 q] for the case described in reference [46]. The authors used a continuous wave pump beam with λp = 405 nm to generate photons with λs,i = 810 nm in a 1.5 mm length bbo crystal. They detected the photons after filtering with ∆λs,i = 10 nm interference filters. The pump beam waist satisfied the condition wp >> L, and the generated photons propagated at ϕs,i = 0◦ . The experiment reported in reference [46] does not show evidence of spacefrequency correlations. In agreement with this fact, the figure shows a lack of correlation for the collinear case. However, under the conditions reported, but in a non collinear configuration, it is not possible to neglect the space and frequency correlations. Finally figure 2.4 (c) plots T r[ˆρ 2 q] for the case described in reference [31]. In this paper, the authors illuminated a 2 mm long bbo crystal with a pump beam with λp = 351.1 nm and wp ≈ 20 µm. Their crystal generated photons at λs,i = 702 nm that propagated at 4o . They collected the photons after using 21
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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 ⎛
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 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 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
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
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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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