Spatial Characterization Of Two-Photon States - GAP-Optique

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4. OAM transfer in noncollinear configurations ing induced by the spatial walk-off. The symmetry breaking implies that the correlations between oam modes do not follow the relationship lp = ls + li. Figure 4.7 also shows that for larger pump beams the azimuthal changes are smoothened out. The insets in figure 4.7 show the oam decomposition at α = 90 ◦ and at α = 360 ◦ . At α = 90 ◦ , the walk-off effect compensates the noncollinear effect, and the weight of the ls = 0 mode is larger than the weight of any other mode. This angle is optimal for the generation of heralded single photons with a Gaussian shape. The degree of spatial entanglement between the photons also exhibits azimuthal variations depending on their emission direction. Figure 4.8 shows the ] as a function of the azimuthal angle α with and with- signal purity T r[ρ2 signal out walk-off, for a pump beam waist wp = 100 µm with collection modes of ws = wi = 50 µm. When considering the walk-off, the degree of entanglement increases and becomes a function of the azimuthal angle. The purity has a minimum for α = 0◦ and α = 180◦ and maximum for α = 90◦ and α = 270◦ . The spatial azimuthal dependence affects especially those experimental configurations where photons from different parts of the cone are used, as in the case in the experiment reported in reference [12] for the generation of photons entangled in polarization. Using two identical type-i spdc crystals with the optical axes rotated 90◦ with respect to each other, the authors generated a space-frequency quantum state given by |Ψ〉 = 1 √ dqsdqi Φ 2 1 q(qs, qi)|H〉s|H〉i + Φ 2 q(qs, qi)|V 〉s|V 〉i (4.8) where Φ 1 q(qs, qi) is the spatial mode function of the photons generated in the first crystal Φ 1 q(qs, qi) = Φq(qs, qi, α = 0 ◦ ) exp [iL tan ρ0(q y s + q y i )], (4.9) and Φ 2 q(qs, qi) is the spatial mode function of the photons generated in the second crystal Φ 2 q(qs, qi) = Φq(qs, qi, α = 90 ◦ ). (4.10) Since the photons generated in the first crystal are affected by the walk-off as they pass by the second crystal, the mode functions are different. Following chapter 2, the polarization state of the generated photons is calculated by tracing out the spatial variables from equation 4.8. The resulting state is described by the density matrix where 46 ˆρp = 1 |H〉s|H〉i〈H|s〈H|i + |V 〉s|V 〉i〈V |s〈V |i 2 +c|H〉s|H〉i〈V |s〈V |i + |V 〉s|V 〉i〈H|s〈H|i (4.11) c = dqsdqiΦ 1 q(qs, qi)Φ 2 q(qs, qi). (4.12)
4.3. Effect of the Poynting vector walk-off on the OAM transfer Length 0.5mm 2mm beam waist Concurrence 1 50 500m Figure 4.9: The concurrence of the polarization entangled state given by equation 4.8 decreases when the effect of the azimuthal variation is stronger, that is for small pump beam waist or for large crystals. The degree of correlation between space and polarization is given by the purity of the polarization state T r[ρ 2 p] = 0.5 0.0 1 + |c|2 . (4.13) 2 If the walk-off effect is negligible T r[ρ 2 p] = 1, then space and polarization are not correlated. If the walk-off is not negligible, the azimuthal changes in the spatial shape are correlated with the polarization of the photons. Figure 4.9 indirectly shows the effect of the azimuthal spatial information over the polarization entanglement. The entanglement between the photons is quantified using the concurrence C = |c|, which is equal to 1 for maximally entangled states. The figure shows the variation of C as a function of the pump beam waist for two crystal lengths. For small pump beam waist, the azimuthal effect is stronger and the concurrence decreases. As the waist increases, the walk-off effect becomes weaker and the value of the concurrence increases. This effect is more important for longer crystals, where the walk-off modifies the pump beam shape more. 4.3.2 Experiment To experimentally corroborate the predicted azimuthal changes in the signal spatial shape we set up the experiment described by the parameters listed in table 4.3 and shown in figure 4.10. We used a continuous wave diode laser emitting at λp = 405 nm and with an approximate Gaussian spatial profile, obtained by using a spatial filter. A half wave plate (hwp) controlled the direction of polarization of the beam, while a lense focalized it on the input face of the crystal, with a beam waist of wp = 136 µm. The 5 mm liio3 crystal produced pairs of photons at 810 nm that propagated at ϕs,i = 4 ◦ . Due to the crystal birefringence, the pump beam exhibited a Pointing vector walk-off 47
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DOCTORAL THESIS IN PHOTONICS ICFO B
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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 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 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.
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
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Abstract La notación matricial int
Page 118 and 119:
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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