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
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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
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2.2. Correlations between space and
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2.3. Correlations between signal an
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2.3. Correlations between signal an
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Signal purity 1 0 (a) 0.1 3 2.3. Co
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CHAPTER 3 Spatial correlations and
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3.2. OAM transfer in general SPDC c
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3.2. OAM transfer in general SPDC c
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Phase front 3.3. OAM transfer in co
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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
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
Page 142 and 143: 2. Correlations and entanglement Fi
Page 144 and 145: 2. Correlations and entanglement q
Page 146 and 147: 2. Correlations and entanglement Si
Page 148 and 149: 2. Correlations and entanglement ri
Page 150 and 151: 3. Spatial correlations and OAM tra
Page 157 and 158: CHAPTER 4 OAM transfer in noncollin
Page 159 and 160: 4.2. Effect of the pump beam waist
Page 161 and 162: 1.0 0.0 4.2. Effect of the pump bea
Page 163 and 164: Coincidences 800 0 -4 4.2. Effect o
Page 165: Weight 1 0 4.3. Effect of the Poynt
Page 169 and 170: Before the crystal 4.3. Effect of t
Page 171 and 172: CHAPTER 5 Spatial correlations in R
Page 173 and 174: x y z 5.1. The quantum state of Sto
Page 175 and 176: where 5.2. Orbital angular momentum
Page 177 and 178: weight 1 0.6 -180º -90º 0º 90º
Page 179: 5.3. Spatial entanglement 5.20 is s
Page 182 and 183: 6. Summary Experiments that explain
Page 184 and 185: A. The matrix form of the mode func
Page 186 and 187: A. The matrix form of the mode func
Page 189 and 190: APPENDIX B Integrals of the matrix
Page 191 and 192: APPENDIX C Methods for OAM measurem
Page 193 and 194: Bibliography [1] M. A. Nielsen and
Page 195 and 196: Bibliography [27] J. P. Torres, A.
Page 197 and 198: Bibliography [55] M. C. Booth, M. A
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