Spatial Characterization Of Two-Photon States - GAP-Optique

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2. Correlations and entanglement Signal purity Filters 1 0 0.1 Collection mode 1mm 0nm 1nm 10nm →∞ Figure 2.6: The correlations between photons can not be suppressed by filtering in only one degree of freedom. The purity of the signal state is only maximum when using ultra-narrow spatial and frequency filters. In this case, the 1 nm filters are broad enough to allow the correlations, and the 10 nm filters are equivalent to infinitely broad ones. The parameters used to generate this figure are listed in table 2.2. The differences between the matrix C and matrix B in the last section, are due to the order of the primed and unprimed variables in the arguments of the mode functions in equations 2.14 and 2.21. The next part of this section uses equation 2.20, to calculate the values of the purity of the signal photon for different spdc configurations. 2.3.3 Numerical calculations Figure 2.6 shows the space-frequency purity of the signal photon as a function of the spatial filter width for different values of the frequency filter width. Consider a degenerate type-i spdc configuration, where a pump beam with wavelength λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength λ 0 s = λ 0 i = 810 nm at an angle ϕs,i = 10 ◦ , and the collection modes for signal and idler are assumed to be equal (ws = wi). All these parameters are listed in the first column of table 2.2. As was discussed in the first part of this section, it is possible to achieve maximal separability between the photons by using infinitely narrow filters in both space and frequency. In the region of small values of ws (= wi) a considerable correlation between signal and idler exists even in the case of infinitely narrow frequency filters. Different values for the signal photon purity can be achieved by changing the filter width, as shown by figure 2.6. Additionally, the figure shows how the purity T r[ˆρ 2 signal ] is confined between the values obtained for ∆λs = ∆λi → 0 nm and ∆λs, ∆λi → ∞. This limits can be tailored by modifying the other parameters of the spdc configuration. Figure 2.7 shows T r[ˆρ 2 signal ] as a function of the pump beam waist wp for 26
Signal purity 1 0 (a) 0.1 3 2.3. Correlations between signal and idler (b) 0.1 Pump waist 3 (c) 0.1 Emission angle Figure 2.7: The effect of the emission angle on the signal purity changes depending on the filters used in space and frequency. When both filters have a finite size like in (a), the purity increases with the angle of emission, and has a maximum for a fixed value of the pump beam waist. When the frequency filters are infinitely narrow like in (b), the dependence of the purity on the angle of emission disappears. When using narrow filters in space, as in (c), the purity is equal to 1 for a certain pump beam waist that changes with the angle of emission. The parameters used to generate this figure are listed in table 2.2. Table 2.2: The parameters used in figures 2.6 and 2.7 Parameter Figure 2.6 Figure 2.7(a) Figure 2.7(b) Figure 2.7(c) Crystal liio3 liio3 liio3 liio3 L 1 mm 1 mm 1 mm 1 mm ρ0 0 ◦ 0 ◦ 0 ◦ 0 ◦ T0 → ∞ → ∞ → ∞ → ∞ wp 400 µm 400 µm 400 µm 400 µm λp 405 nm 405 nm 405 nm 405 nm ∆λp → 0 → 0 → 0 → 0 ws variable → ∞ 400 µm 400 µm λ 0 s 810 nm 810 nm 810 nm 810 nm ∆λs 0, 1, 10 nm → ∞ 10 nm → 0 10 nm ϕs 10 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦ 5, 10, 20 ◦ various values of the emission angle ϕs = ϕi, in degenerate type-i spdc configurations described by the parameters listed in the second, third and fourth columns of table 2.2. Figure 2.7 (a) shows the case of finite spatial and temporal filters. The purity of the signal photon is always smaller than one, increases with the emission angle, and has a maximum for a given value of the pump beam waist. For a very narrow band frequency filter with a ws = wi = 400 µm spatial filter, figure 2.7 (b) shows how the correlation between the photons is minimal for small pump beams. The emission angle for this particular crystal length is irrelevant. Finally, figure 2.7 (c) considers a frequency filter ∆λs = ∆λi = 10 nm with infinitely narrow spatial filters. In this case, maximal purity appears for each emission angle at a particular value of the pump beam waist. Conclusion <strong>Two</strong> photons with two degrees of freedom compose the two-photon state generated in spdc. The correlations between all four elements affect the state of single photons or two-photon states with one degree of freedom. Using the pu- 3mm 5º 10º 20º 27
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 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: 2.3. Correlations between signal an
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 92 and 93: C. Methods for OAM measurements Inp
Page 94 and 95: Bibliography [13] M. Barbieri, C. C
Page 96 and 97:
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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