Spatial Characterization Of Two-Photon States - GAP-Optique

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When he [Kepler] found that his long cherished beliefs did not agree with the most precise observations, he accepted the uncomfortable facts, he preferred the hard truth to his dearest illusions. That is the heart of science. Cosmos - Carl Sagan
Abstract In the same way that electronics is based on measuring and controlling the state of electrons, the technological applications of quantum optics will be based on our ability to generate and characterize photonic states. The generation of photonic states is traditionally associated to nonlinear optics, where the interaction of a beam and a nonlinear material results in the generation of multi-photon states. The most common process is spontaneous parametric down-conversion (spdc), which is used as a source of pairs of photons not only for quantum optics applications but also for quantum information and quantum cryptography [1, 2]. The popularity of spdc lies in the relative simplicity of its experimental realization, and in the variety of quantum features that down-converted photons exhibit. For instance, a pair of photons generated via spdc can be entangled in polarization [3, 4], frequency [5, 6], or in the equivalent degrees of freedom of orbital angular momentum, space, and transverse momentum [7, 8, 9, 10, 11]. Standard spdc applications focus on a single degree of freedom, wasting the entanglement in other degrees of freedom and the correlations between them. Among the few configurations using more than one degree of freedom are hyperentanglement [12, 13], spatial entanglement distillation using polarization [14], or control of the joined spectrum using the pump’s spatial properties [6]. This thesis describes the spatial properties of the two-photon state generated via spdc, considering the different parameters of the process, and the correlations between space and frequency. To achieve this goal, I use the purity to quantify the correlations between the photons, and between the degrees of freedom. Additionally, I study the spatial correlations by describing the transfer of orbital angular momentum (oam) from the pump to the signal and idler photons, taking into account the pump, the detection system and other parameters of the process. This thesis is composed of six chapters. Chapter 1 introduces the mode function, used throughout the thesis to describe the two-photon state in space and frequency. Chapter 2 describes the correlations between degrees of freedom or photons in the two-photon state, using the purity to quantify such correlations. Chapter 3 explains the mechanism of the oam transfer from pump to signal and idler photons. Chapter 4 describes the effect of different spdc parameters on the oam transfer in noncollinear configurations, both theoretically and experimentally. In analogy with the downconverted case, chapter 5 discusses the two-photon state generated via Raman transition, by describing its mode function, the correlations between different parts of the state, and the oam transfer in the process. Finally, chapter 6 summarizes the main results presented by this thesis. xi
Page 1: DOCTORAL THESIS IN PHOTONICS ICFO B
Page 5: Spatial Characterization Of Two-Pho
Page 9 and 10: Contents Contents vii Acknowledgeme
Page 11: Acknowledgements This thesis compil
Page 15 and 16: Resumen De la misma manera que la e
Page 17 and 18: Introduction The role of photons in
Page 19: This thesis is based on the followi
Page 22 and 23: 1. General description of two-photo
Page 35 and 36: CHAPTER 2 Correlations and entangle
Page 37 and 38: 2.1. The purity as a correlation in
Page 39 and 40: 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 60 and 61: 4. OAM transfer in noncollinear con
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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
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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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Spatial Characterization Of Two-Photon States - GAP-Optique

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