Spatial Characterization Of Two-Photon States - GAP-Optique

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5. Spatial correlations in Raman transitions with F = 4AB + (A + B) w2 g A + B + w 2 g G = 4CD + (C + D) w2 g C + D + w2 . (5.17) g As in the previous chapter, this mode function can be written as a superposition of spherical harmonics Φs (qs, θs) = (2π) −1/2 ls als (qs) exp (ilsθs) . (5.18) The probability of having a Stokes photon with oam equal to ls is given by the weight of each spiral mode F + G C2ls = (F G)1/2 dqsqs exp − q 4 2 s I 2 G − F ls q 8 2 s (5.19) where Ils are the Bessel functions of the second kind. Only even modes appear in the distribution as a consequence of the symmetry of the Stokes mode function. Figure 5.3 shows the weight of the mode ls = 0 as a function of the angle of emission for different values of the length of the cloud of atoms. For nearly collinear configurations, the probability of having a Gaussian Stokes photon is one, therefore, in this case the process satisfies the relation lp+lc = ls +las. For these kind of configurations, A = D in equation 5.12, and therefore the Stokes spatial mode function shows cylindrical symmetry in the transverse planes (xs, ys) and (xas, yas). This is the case in most experimental configurations [59, 64, 68], and in fact, reference [23] experimentally proves the relationship lp + lc = ls + las for collinear configurations. The probability of having a Gaussian Stokes photon decreases as other modes appear in the distribution in highly noncollinear configurations. The length of the cloud controls the importance of the other modes and it is possible to obtain a spatial mode with cylindrical symmetry for any emission angle, by fulfilling the condition L = R 1 + 2R2 w2 −1/2 . (5.20) p This condition reduces to L = R, when the beam waist is much larger than the transverse size of the cloud. Any deviation from a spherical volume of interaction (described by this condition) introduces ellipticity in the mode function. For this reason, a highly elliptical configuration, as the one described in reference [60], will not satisfy the oam selection rule. Figure 5.4 shows the weight of the mode ls = 0 as a function of the length of the cloud for different values of the emission angle. For any angle, the probability of having a Gaussian mode is maximum at the length given by equation 5.20. In a collinear configuration, the Stokes photon always has a Gaussian distribution, independently of the length of the cloud. As the angle of emission increases, the length of the cloud becomes more important. The mode weight is weakly affected by the change of the cloud length when the length is much longer than the other relevant parameters: ws, wg and R. This is especially evident for an angle of emission ϕ = 90◦ . 56
weight 1 0.6 -180º -90º 0º 90º 180º 5.3. Spatial entanglement length 0.2mm 0.4mm 1mm 2mm angle Figure 5.3: The weight of the Gaussian mode in the oam decomposition for the Stokes photon has a maximum in the collinear configuration. In noncollinear configurations the probability of a Gaussian Stokes photon decreases as the length of the cloud increases. Table 5.1 lists the parameters used to generate this figure. weight angle 1 0.6 0.2 Length 1.5 mm Figure 5.4: The probability of having a Gaussian Stokes photon is maximum in a collinear configuration independently of the cloud length. In the noncollinear cases the maximum appears at the length given by equation 5.20, in these configurations the probability of having a Gaussian Stokes photon changes drastically with the cloud length and the emission angle. Table 5.1 lists the parameters used to generate this figure. 5.3 Spatial entanglement The spatial correlations between the generated photons inherit the angular dependence from the Stokes oam content. To explore this phenomena, this 0º 10º 90º 57
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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
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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
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
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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 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: where 5.2. Orbital angular momentum
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
Page 195 and 196: Bibliography [27] J. P. Torres, A.
Page 197 and 198: Bibliography [55] M. C. Booth, M. A
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Spatial Characterization Of Two-Photon States - GAP-Optique

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