Spatial Characterization Of Two-Photon States - GAP-Optique

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C. Methods for OAM measurements Input hologram lens Gaussian mode non-Gaussian mode Figure C.1: A hologram that records the interference of a lg mode with a Gaussian beam, can be used to detect that particular lg mode. Only if the generating lg mode is incident on the hologram, a Gaussian beam is recovered. If another mode is incident, the hologram will generate non-Gaussian beams. Single mode optical fibers distinguish between Gaussian and non-Gaussian beams. Input beam splitter Dove prism mirror even modes odd modes Figure C.2: <strong>Two</strong> Dove prisms rotated with respect to each other by an angle π/2 induce a relative rotation of π between the two arms of a Mach-Zender interferometer. As a consequence of the rotation, odd and even lg modes leave the interferometer at different outputs after the second beam splitter. Reference [15] introduces this principle as a base of an oam sorter. 72
Bibliography [1] M. A. Nielsen and I. L. Chuang, Quantum computation and quantum information. Cambridge, UK: Cambridge University Press, 2000. [2] M. Fox, Quantum Optics, An Introduction. Oxford, UK: Oxford University Press, 2006. [3] S. P. Walborn, S. Pádua, and C. H. Monken, “Hyperentanglement-assisted bell-state analysis,” Phys. Rev. A, vol. 68, p. 042313, 2003. [4] P. G. Kwiat, P. H. Eberhard, A. M. Steinberg, and R. Y. Chiao, “Proposal for a loophole-free bell inequality experiment,” Phys. Rev. A, vol. 49, pp. 3209–3220, 1994. [5] P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett., vol. 100, p. 133601, 2008. [6] A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett., vol. 99, p. 243601, 2007. [7] A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, “Entanglement of the orbital angular momentum states of photons,” Nature, vol. 412, pp. 313– 316, 2001. [8] S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “<strong>Two</strong>photon entanglement of orbital angular momentum states,” Phys. Rev. A, vol. 65, p. 033823, 2002. [9] G. Molina-Terriza, J. P. Torres, and L. Torner, “Twisted photons,” Nature Phys., vol. 3, p. 305, 2007. [10] S. P. Walborn and C. H. Monken, “Transverse spatial entanglement in parametric down-conversion,” Phys. Rev. A, vol. 76, p. 062305, 2007. [11] C. K. Law and J. H. Eberly, “Analysis and interpretation of high transverse entanglement in optical parametric down conversion,” Phys. Rev. Lett., vol. 92, p. 127903, 2004. [12] J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, “Generation of hyperentangled photon pairs,” Phys. Rev. Lett., vol. 95, p. 260501, 2005. 73
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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
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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
Page 142 and 143: 2. Correlations and entanglement Fi
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Page 157 and 158: CHAPTER 4 OAM transfer in noncollin
Page 159 and 160: 4.2. Effect of the pump beam waist
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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: APPENDIX C Methods for OAM measurem
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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