Spatial Characterization Of Two-Photon States - GAP-Optique

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1. General description of two-photon states When the amplitude of the electric field increases, the higher orders term in equation 1.2 become relevant, and then a nonlinear response of the material to the field appears. Optical nonlinear phenomena resulting from this kind of interaction include the generation of harmonics, the Kerr effect, Raman scattering, self-phase modulation, and cross-phase modulation [34]. Spontaneous parametric down-conversion, and other second order nonlinear processes, result from the second order polarization, defined as the first nonlinear term in the polarization tensor P (2) = ɛ0χ (2) EE. (1.4) The quantization of the electromagnetic field leads to a quantization of the second order polarization, so that the nonlinear polarization operator ˆ P (2) becomes ˆP (2) = ɛ0χ (2) ( Ê(+) + Ê(−) )( Ê(+) + Ê(−) ), (1.5) where Ê(+) and Ê(−) are the positive and negative frequency parts of the field operator [36]. The positive frequency part of the electric field operator is a function of the annihilation operator â(k), and is defined at position rn xnˆxn + ynˆyn + znˆzn and time t as = Ê (+) 1/2 ωn n (rn, t) = ien 2ɛ0v dkn exp [ikn · rn − iωnt]â(kn), (1.6) where the magnitude of the wave vector k satisfies k 2 = ωn/c. The volume v contains the field, and en is the unitary polarization vector. The negative frequency part of the field is the Hermitian conjugate of the positive part, Ê (−) n (rn, t) = Ê(+)† n (rn, t). Thus, the negative frequency part is a function of the creation operator â † (k). Following references [37] and [38], in first order perturbation theory, the interaction of a volume V of a material with a nonlinear polarization ˆ P (2) and the field Êp(rp, t), produces a system described by the state |ΨT 〉 ∝ |1〉p|0〉s|0〉i − i τ 0 dt ˆ HI|1〉p|0〉s|0〉i, (1.7) where τ is the interaction time, |1〉p|0〉s|0〉i is the initial state of the field, and the interaction Hamiltonian reads ˆHI = − dV ˆ P (2) Êp(rp, t). (1.8) V The first term at the right side of equation 1.7 describes a one photon system, while the second term describes a two-photon system. In what follows we will consider only the second term as we are mainly interested in the generation of pairs of photons. The two-photon system state is given by |Ψ〉 = i τ 0 dt ˆ HI|1〉p|0〉s|0〉i, (1.9) or ˆρ = |Ψ〉〈Ψ| in the density matrix formalism. Seven of the eight terms that compose the interaction Hamiltonian vanish when they are applied over the state |1〉p|0〉s|0〉i. The remaining term is proportional to the annihilation 4
1.3. Approximations and other considerations operator in mode p, and the creation operators in modes s and i. Assuming constant χ (2) , the two-photon state reduces to (2) τ iɛ0χ |Ψ〉 = dt dV 0 V Ê(+) p (rp, ωp) Ê(−) s (rs, ωs) Ê(−) i (ri, ωi)|1〉p|0〉s|0〉i, (1.10) which can be written as |Ψ〉 ∝ dks dkiΦ(ks, ωs, ki, ωi)a † (ks)a † (ki)|0〉s|0〉i; (1.11) where Φ(ks, ωs, ki, ωi) is known as the mode function and, assuming that the pump has certain spatial distribution Ep(kp), is given by τ Φ(ks, ωs, ki, ωi) ∝ dt dV dkpEp(kp) 0 V × exp [ikp · rp − iks · rs − iki · ri − i(ωp − ωs − ωi)t] × â(kp)|1〉p. (1.12) The mode function contains all the information about the generated two-photon system in space and frequency, not only about their individual state but about their correlations. This function is therefore highly complex, and to calculate any feature of the two-photon state it is necessary to simplify it, as the next section shows. 1.3 Approximations and other considerations Analytical calculations of any features of the down converted photons require simplification of the mode function in equation 1.12. This section lists the most important approximations used in this thesis, as well as the restrictions that they impose. Separation of transversal and longitudinal components A field with a wave vector k almost parallel to its propagation direction spreads only a little in the transversal direction. It is then possible, to consider the longitudinal and transversal components of the wave vector separately, k = kzˆz + q. (1.13) As the wave vector’s magnitude k = ωn/c is bigger than the transversal component’s magnitude q = |q| = (k 2 x + k 2 y) 1/2 , the magnitude of the longitudinal component kz is always a real number, kz = k 2 1 − q 2 2 . (1.14) Assuming that the pump, the signal, and the idler are such fields, the mode function becomes τ Φ(qs, ωs, qi, ωi) ∝ 0 dt dV V dqpEp(qp) exp [ik z pzp − ik z szs − ik z i zi] × exp [iq p · r ⊥ p − iq s · r ⊥ s − iq i · r ⊥ i − i(ωp − ωs − ωi)t] × â(q p)|1〉p. (1.15) where r ⊥ n = xnˆxn + ynˆyn is the transversal position vector. 5
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 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
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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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