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
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 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
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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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