Spatial Characterization Of Two-Photon States - GAP-Optique

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2. Correlations and entanglement chosen. In the particular case of a two-photon system, the purity of the spatial part can be used to study correlations between the degrees of freedom, or the purity of the signal photon can be used to study the entanglement between the photons. The next sections explore both these approaches. 2.2 Correlations between space and frequency In type-i spdc, the photons generated have a polarization orthogonal to the polarization of the incident beam. Therefore, the two-photon state generated in a type-i process has only two degrees of freedom: frequency and transversal spatial distribution. In this section, I study the origin and the characteristics of the correlations between those degrees of freedom by using the purity of the spatial part as a correlation indicator. Except for hyperentanglement configurations [12, 13] and some other novel configurations [6, 14], most applications of type-i spdc processes use just one of the degrees of freedom, ignoring the correlation that may exist between space and frequency. That is, those configurations assume a high degree of spatial purity. This section explores the conditions in which this assumption is valid. The first part of this section contains a discussion about the origin and suppression of the correlations. The second part includes the calculations for the spatial purity that is used as correlation indicator. The third part shows the results of numerical calculation of the spatial purity in several cases. 2.2.1 Origin of the correlations According to the two-photon mode function, equation 1.34, the correlations between space and frequency appear as cross terms in the variables associated to those degrees of freedom, as figure 2.2 shows. The correlations between q s x q s y q i x y qi s i qs x a h i j k l q s y h b m n p r qi y i m c s t u Figure 2.2: Cross terms in matrix A generate correlations between space and frequency. Without those terms, the matrix A is composed by two nonzero block matrices, one for each degree of freedom. space and time are always relevant, if at least one of the cross terms k, l, p, r, t, u, v, w is comparable with another term of matrix A. Therefore, it is possible to suppress the correlations by making all the cross terms negligible, or by increasing the values of the other terms. The use of ultra-narrow filters in one of the degrees of freedom will suppress most of the information available in it, destroying any possible correlation with 18 q i x j n s d v w s k p t v f z i l r u w z g
2.2. Correlations between space and frequency other degrees of freedom. Filtering makes the diagonal terms of matrix A larger than the rest of the terms, including the terms responsible for the correlations. But while they remove the correlations, the filters reduce the amount of photons available for any measurement. An appropriate design of the source makes it possible to suppress cross terms without using filters. Such a design should fulfill the conditions Np − Ns cos ϕs Np − Ni cos ϕi Np − Ns cos ϕs Np − Ni cos ϕi Np − Ns cos ϕs = 0 Np − Ni cos ϕi = 0 1 + w2 p γ2L2 1 + w2 p γ 2 L 2 1 − w2 p γ 2 L 2 1 − w2 p γ 2 L 2 Ns sin ϕs tan ϕi Ni sin ϕi tan ϕs = 0 = 0 = 0 = 0. (2.8) These conditions can be met for example by using a long crystal, a highly focused pump beam, for which wp/γL → 0, and emission angles such that cos ϕs = Np/Ns and cos ϕi = Np/Ni. Even though these conditions have been deduced in a simplified regime, reference [45] shows that these conditions are sufficient to remove the correlations when considering the general mode function of equation 1.12. Once the correlations are suppressed, the two-photon mode function can be written as a product of a spatial and a frequency two-photon mode function like Φ(q s, ωs, q i, ωi) = Φq(q s, q i)ΦΩ(Ωs, Ωi). (2.9) Since the global two-photon system in space and frequency is in a pure state, the presence of correlations can be confirmed by using the purity. The next section calculates the purity of the spatial state, which can be used as an indicator of correlations between space and frequency. 2.2.2 <strong>Two</strong>-photon spatial state To study space/frequency correlations I will describe the purity of the spatial part of the two-photon state. As the purity of the spatial and frequency parts of the state are equal, all results derived for the spatial two-photon state extend to a frequency two-photon state. After tracing-out the frequency from the two-photon state, the remaining state describes the spatial part of the state. The reduced density matrix for 19
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 and 12: Acknowledgements This thesis compil
Page 13 and 14: Abstract In the same way that elect
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: 2.1. The purity as a correlation in
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 ⎛
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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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