Abstract

We experimentally demonstrate that paired photons generated in different sections of a down-conversion cone, when some of the interacting waves show Poynting vector walk-off, carry different spatial correlations, and therefore a different degree of spatial entanglement. This is shown to be in agreement with theoretical results. We also discuss how this azimuthal distinguishing information of the down-conversion cone is relevant for the implementation of quantum sources aimed at the generation of entanglement in other degrees of freedom, such as polarization.

© 2007 Optical Society of America

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References

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  1. G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007).
    [CrossRef]
  2. A. Vaziri, J. Pan, T. Jennewein, G. Weihs and A. Zeilinger, "Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum," Phys. Rev. Lett. 91, 227902 (2003).
    [CrossRef] [PubMed]
  3. G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, " Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
    [CrossRef] [PubMed]
  4. J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
    [CrossRef]
  5. J. P. Torres, A. Alexandrescu and L. Torner, "Quantum spiral bandwidth of entangled two-photon states," Phys. Rev. A 68050301(R) (2003).
    [CrossRef]
  6. C. K. Law and J. H. Eberly, "Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion," Phys. Rev. Lett. 92, 127903 (2004).
    [CrossRef] [PubMed]
  7. H. H. Arnaut and G. A. Barbosa, "Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs of Photons Generated by Parametric Down-Conversion," Phys. Rev. Lett. 85, 286 (2000).
    [CrossRef] [PubMed]
  8. A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
    [CrossRef] [PubMed]
  9. P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, "How focused pumping affects type-II spontaneous parametric down-conversion," Phys. Rev. A 72, 033803 (2005).
    [CrossRef]
  10. P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999).
    [CrossRef]
  11. J. Altepeter, E. Jeffrey, and P. Kwiat, "Phase-compensated ultra-bright source of entangled photons," Opt. Express,  13, 8951-8959(2005).
    [CrossRef] [PubMed]
  12. J. P. Torres, G. Molina-Terriza and L. Torner, "The spatial shape of entangled photon states generated in noncollinear, walking parametric downconversion," J. Opt. B: Quantum Semiclass. Opt. 7, 235-239 (2005).
    [CrossRef]
  13. A. Migdall, "Polarization directions of noncollinear phase-matched optical parametric downconversion output," J. Opt. Soc. Am. B,  141093-1098 (1997).
    [CrossRef]
  14. M. H. Rubin, "Transverse correlation in optical spontaneous parametric down-conversion," Phys. Rev. A 54, 5349 (1996).
    [CrossRef] [PubMed]
  15. G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych and J. P. Torres, "Control of the shape of the spatial mode function of photons generated in noncollinear spontaneous parametric down-conversion," Phys. Rev. A 72, 065802 (2005).
    [CrossRef]
  16. M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe and S. P. Kulik, "Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 99, 063901 (2007).
    [CrossRef] [PubMed]
  17. G. Molina-Terriza, J. P. Torres and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
    [CrossRef] [PubMed]
  18. M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, "Effect of spatial filtering on the Schmidt decomposition of entangled photons," Phys. Rev. A 74, 012309 (2006).
    [CrossRef]

2007 (2)

G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007).
[CrossRef]

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe and S. P. Kulik, "Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 99, 063901 (2007).
[CrossRef] [PubMed]

2006 (1)

M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, "Effect of spatial filtering on the Schmidt decomposition of entangled photons," Phys. Rev. A 74, 012309 (2006).
[CrossRef]

2005 (6)

G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych and J. P. Torres, "Control of the shape of the spatial mode function of photons generated in noncollinear spontaneous parametric down-conversion," Phys. Rev. A 72, 065802 (2005).
[CrossRef]

J. Altepeter, E. Jeffrey, and P. Kwiat, "Phase-compensated ultra-bright source of entangled photons," Opt. Express,  13, 8951-8959(2005).
[CrossRef] [PubMed]

J. P. Torres, G. Molina-Terriza and L. Torner, "The spatial shape of entangled photon states generated in noncollinear, walking parametric downconversion," J. Opt. B: Quantum Semiclass. Opt. 7, 235-239 (2005).
[CrossRef]

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, " Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, "How focused pumping affects type-II spontaneous parametric down-conversion," Phys. Rev. A 72, 033803 (2005).
[CrossRef]

2004 (1)

C. K. Law and J. H. Eberly, "Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion," Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef] [PubMed]

2003 (2)

J. P. Torres, A. Alexandrescu and L. Torner, "Quantum spiral bandwidth of entangled two-photon states," Phys. Rev. A 68050301(R) (2003).
[CrossRef]

A. Vaziri, J. Pan, T. Jennewein, G. Weihs and A. Zeilinger, "Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum," Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

2002 (1)

G. Molina-Terriza, J. P. Torres and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

2001 (1)

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

2000 (1)

H. H. Arnaut and G. A. Barbosa, "Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs of Photons Generated by Parametric Down-Conversion," Phys. Rev. Lett. 85, 286 (2000).
[CrossRef] [PubMed]

1999 (1)

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999).
[CrossRef]

1997 (1)

1996 (1)

M. H. Rubin, "Transverse correlation in optical spontaneous parametric down-conversion," Phys. Rev. A 54, 5349 (1996).
[CrossRef] [PubMed]

J. Opt. B: Quantum Semiclass. Opt. (1)

J. P. Torres, G. Molina-Terriza and L. Torner, "The spatial shape of entangled photon states generated in noncollinear, walking parametric downconversion," J. Opt. B: Quantum Semiclass. Opt. 7, 235-239 (2005).
[CrossRef]

J. Opt. Soc. Am. B (1)

Nature (1)

A. Mair, A. Vaziri, G. Weihs and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature 412, 313-316 (2001).
[CrossRef] [PubMed]

Nature Phys. (1)

G. Molina-Terriza, J. P. Torres and L. Torner, "Twisted photons," Nature Phys. 3, 305 - 310 (2007).
[CrossRef]

Opt. Express (1)

Phys. Rev. A (6)

M. P. van Exter, A. Aiello, S. S. R. Oemrawsingh, G. Nienhuis, and J. P. Woerdman, "Effect of spatial filtering on the Schmidt decomposition of entangled photons," Phys. Rev. A 74, 012309 (2006).
[CrossRef]

M. H. Rubin, "Transverse correlation in optical spontaneous parametric down-conversion," Phys. Rev. A 54, 5349 (1996).
[CrossRef] [PubMed]

G. Molina-Terriza, S. Minardi, Y. Deyanova, C. I. Osorio, M. Hendrych and J. P. Torres, "Control of the shape of the spatial mode function of photons generated in noncollinear spontaneous parametric down-conversion," Phys. Rev. A 72, 065802 (2005).
[CrossRef]

P. S. K. Lee, M. P. van Exter, and J. P. Woerdman, "How focused pumping affects type-II spontaneous parametric down-conversion," Phys. Rev. A 72, 033803 (2005).
[CrossRef]

P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum and P. H. Eberhard, "Ultrabright source of polarization-entangled photons," Phys. Rev. A 60, R773 (1999).
[CrossRef]

J. P. Torres, A. Alexandrescu and L. Torner, "Quantum spiral bandwidth of entangled two-photon states," Phys. Rev. A 68050301(R) (2003).
[CrossRef]

Phys. Rev. Lett. (7)

C. K. Law and J. H. Eberly, "Analysis and Interpretation of High Transverse Entanglement in Optical Parametric Down Conversion," Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef] [PubMed]

H. H. Arnaut and G. A. Barbosa, "Orbital and Intrinsic Angular Momentum of Single Photons and Entangled Pairs of Photons Generated by Parametric Down-Conversion," Phys. Rev. Lett. 85, 286 (2000).
[CrossRef] [PubMed]

A. Vaziri, J. Pan, T. Jennewein, G. Weihs and A. Zeilinger, "Concentration of Higher Dimensional Entanglement: Qutrits of Photon Orbital Angular Momentum," Phys. Rev. Lett. 91, 227902 (2003).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, R. Ursin, and A. Zeilinger, " Experimental Quantum Coin Tossing," Phys. Rev. Lett. 94, 040501 (2005).
[CrossRef] [PubMed]

J. T. Barreiro, N. K. Langford, N. A. Peters, and P. G. Kwiat, "Generation of Hyperentangled Photon Pairs," Phys. Rev. Lett. 95, 260501 (2005).
[CrossRef]

M. V. Fedorov, M. A. Efremov, P. A. Volkov, E. V. Moreva, S. S. Straupe and S. P. Kulik, "Anisotropically and High Entanglement of Biphoton States Generated in Spontaneous Parametric Down-Conversion," Phys. Rev. Lett. 99, 063901 (2007).
[CrossRef] [PubMed]

G. Molina-Terriza, J. P. Torres and L. Torner, "Management of the Angular Momentum of Light: Preparation of Photons in Multidimensional Vector States of Angular Momentum," Phys. Rev. Lett. 88, 013601 (2002).
[CrossRef] [PubMed]

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1.

(a) Diagram of the experimental set up and (b) The down-conversion cone. Single photon detectors are located in opposite sides of the cone, forming an angle α with the YZ plane.

Fig. 2.
Fig. 2.

Images showing the spatial shape of the mode function of the signal photon when measuring coincidences rates. Upper row corresponds to theoretical predictions, and the lower row corresponds to experimental data. (a) α= 0°; (b) α= 90°; (c) α= 180° and (d) α= 270°. See also the corresponding movie. [Media 1]

Fig. 3.
Fig. 3.

Weight of the OAM modes ls = 0 (solid line), and all other modes (dashed lines) as a function of the angle α. (a), (c) and (d) w 0 = 100μm; (b), (e) and (f) w 0 = 600μm. (c) and (e) show the OAM distribution for α = 0°, and (d) and (f) corresponds to α = 90°. We assume negligible spatial filtering (ws ≃ 0). Dot-dashed lines: no spatial walk-off.

Fig. 4.
Fig. 4.

(a) Schmidt number (K) as a function of the angle α. The width of the collection mode is ws = 50μm. (b) Schmidt number as a function of the width of the collection mode for different values of α. In all cases, the pump beam width is w 0 = 100μm. The Schmidt number for the case with negligible walk-off (dashed lines) is shown for comparison.

Fig. 5.
Fig. 5.

Concurrence (C) of the polarization entangled bi-photon state generated in a two crystal configuration, as a function of the pump beam, for two different values of the crystal length. The non-collinear angle is φ = 2°, and the pump beam waist is w 0 = 100μm.

Equations (5)

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Φ ( p , q ) = N exp { ( Γ L ) 2 4 Δ k 2 + i Δ k L 2 }
× exp { ( p x + q x ) 2 w 0 2 + ( p y + q y ) 2 w 0 2 cos 2 φ 4 }
× exp { p 2 w s 2 4 q 2 w s 2 4 }
Ψ = 1 2 d p d q [ Φ 1 ( p , q ) H , p s H , q i + Φ 2 ( p , q ) V , p s V , q i ]
ρ p = 1 2 { H s H i H s H i + V s V i V s V i + ξ [ H s H i V s V i + V s V i H s H i ] }

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