Abstract

We calculate the coincidence count probabilities when the photon pairs entangled in orbital angular momentum generated via spontaneous parametric downconversion are measured by using holograms with an m-fold dislocation, considering the nonzero crystal length. We find that the coincidence probabilities related to azimuthal changes in hologram positions deviate significantly from a sinusoidal curve, which has often been assumed in simple analyses. It is found that the main cause of this effect is the high-dimensional entanglement in Laguerre–Gaussian modes. We also find that the crystal length causes the hologram positions to change for the maximum coincidence probabilities.

© 2009 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100, 173902 (2008).
    [CrossRef] [PubMed]
  5. D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418-4421 (2000).
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    [CrossRef] [PubMed]
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    [CrossRef]
  24. R. S. Bennink, S. J. Bentley, and R. W. Boyd, “'Two-photon' coincidence imaging with a classical source,” Phys. Rev. Lett. 9, 113601 (2002).
    [CrossRef]
  25. S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system,” Appl. Phys. Lett. 74, 1063 (1999).
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  26. J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multi-photon counting using visible light photon counter,” Appl. Phys. Lett. 74, 902 (1999).
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  27. 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]
  28. A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
    [CrossRef]
  29. C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123-3126 (1998).
    [CrossRef]
  30. S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003).
    [CrossRef] [PubMed]
  31. For the cases where the Rayleigh range of the pump is smaller than the crystal length, L can be considered the Rayleigh range.
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    [CrossRef]
  33. 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]
  34. 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]
  35. This approximation is introduced to simplify the form of the mode function in position representation, which can be easily calculated numerically.
  36. J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
    [CrossRef]
  37. These parameters were determined for various experimental conditions.
  38. D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.
  39. Note that for ∣m∣>1 there are more than one values of the distance d from the dislocation to the optical axis for which the minimum coincidence probability becomes 0. For the plots in Fig. , we selected the smallest values (d/w0=0.35 for m=2 and 0.26 for m=3) that meet this criterion. We have confirmed that the shape of the plots are almost the same even when we select other values of d for which the minimum coincidence becomes 0.
  40. One can easily confirm this fact by using a simple model assuming the state of the source is given by ∣Ψ⟩=α∣0⟩I∣0⟩S+β∣1⟩I∣−1⟩S and the detection basis states are ∣ψ⟩S=sinδS∣0⟩S+cosδS∣−1⟩S and ∣ψ⟩I=sinδI∣0⟩I+cosδI∣1⟩I.
  41. T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
    [CrossRef] [PubMed]

2008 (3)

G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100, 173902 (2008).
[CrossRef] [PubMed]

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre Gauss modes using Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[CrossRef] [PubMed]

A. Ling, A. Lamas-Linares, and C. Kurtsiefer, “Absolute emission rates of spontaneous parametric down-conversion into single transverse Gaussian modes,” Phys. Rev. A 77, 043834 (2008).
[CrossRef]

2007 (4)

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007).
[CrossRef]

M. Stütz, S. Gröblachera, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
[CrossRef] [PubMed]

2006 (3)

R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

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]

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

2005 (4)

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[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]

H. Nihira and C. R. Stroud, Jr., “Robust multipartite multilevel quantum protocols,” Phys. Rev. A 72, 022337 (2005).
[CrossRef]

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]

2004 (4)

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

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]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

2003 (5)

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003).
[CrossRef]

M. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777-785 (2003).
[CrossRef]

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[CrossRef]

A. Vaziri, J. W. 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]

S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003).
[CrossRef] [PubMed]

2002 (6)

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[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]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033823 (2002).
[CrossRef]

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “'Two-photon' coincidence imaging with a classical source,” Phys. Rev. Lett. 9, 113601 (2002).
[CrossRef]

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)

D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418-4421 (2000).
[CrossRef] [PubMed]

1999 (2)

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system,” Appl. Phys. Lett. 74, 1063 (1999).
[CrossRef]

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multi-photon counting using visible light photon counter,” Appl. Phys. Lett. 74, 902 (1999).
[CrossRef]

1998 (1)

C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123-3126 (1998).
[CrossRef]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Aiello, A.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

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]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Alexandrescu, A.

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003).
[CrossRef]

Allen, L.

M. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777-785 (2003).
[CrossRef]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033823 (2002).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Barnett, S.

M. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777-785 (2003).
[CrossRef]

Barnett, S. M.

R. Zambrini and S. M. Barnett, “Quasi-intrinsic angular momentum and the measurement of its spectrum,” Phys. Rev. Lett. 96, 113901 (2006).
[CrossRef] [PubMed]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Barreiro, J. T.

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]

Bartlett, S. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Bennink, R. S.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “'Two-photon' coincidence imaging with a classical source,” Phys. Rev. Lett. 9, 113601 (2002).
[CrossRef]

Bentley, S. J.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “'Two-photon' coincidence imaging with a classical source,” Phys. Rev. Lett. 9, 113601 (2002).
[CrossRef]

Bourennane, M.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Boyd, R. W.

R. S. Bennink, S. J. Bentley, and R. W. Boyd, “'Two-photon' coincidence imaging with a classical source,” Phys. Rev. Lett. 9, 113601 (2002).
[CrossRef]

Bramon, A.

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007).
[CrossRef]

Calvo, G. F.

G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100, 173902 (2008).
[CrossRef] [PubMed]

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007).
[CrossRef]

Ceré, A.

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Cerf, N. J.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Courtial, J.

M. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777-785 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Dalton, R. B.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

de Jong, J. A.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

de Oliveira, A. N.

S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003).
[CrossRef] [PubMed]

Deyanova, Y.

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. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[CrossRef]

Eberly, J. H.

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]

Eliel, E. R.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Franke-Arnold, S.

M. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777-785 (2003).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033823 (2002).
[CrossRef]

Gilchrist, A.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Gisin, N.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Gnacinski, P.

D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418-4421 (2000).
[CrossRef] [PubMed]

Gröblachera, S.

M. Stütz, S. Gröblachera, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

Harvey, M. D.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Hendrych, M.

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]

Hogue, H. H.

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multi-photon counting using visible light photon counter,” Appl. Phys. Lett. 74, 902 (1999).
[CrossRef]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system,” Appl. Phys. Lett. 74, 1063 (1999).
[CrossRef]

Hradil, Z.

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

Jennewein, T.

M. Stütz, S. Gröblachera, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

A. Vaziri, J. W. 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]

Karlsson, A.

N. J. Cerf, M. Bourennane, A. Karlsson, and N. Gisin, “Security of quantum key distribution using d-level systems,” Phys. Rev. Lett. 88, 127902 (2002).
[CrossRef] [PubMed]

Kaszlikowski, D.

D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418-4421 (2000).
[CrossRef] [PubMed]

Kawase, D.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre Gauss modes using Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[CrossRef] [PubMed]

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Kim, J.

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multi-photon counting using visible light photon counter,” Appl. Phys. Lett. 74, 902 (1999).
[CrossRef]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system,” Appl. Phys. Lett. 74, 1063 (1999).
[CrossRef]

Kurtsiefer, C.

A. Ling, A. Lamas-Linares, and C. Kurtsiefer, “Absolute emission rates of spontaneous parametric down-conversion into single transverse Gaussian modes,” Phys. Rev. A 77, 043834 (2008).
[CrossRef]

Kwiat, P. G.

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]

Lamas-Linares, A.

A. Ling, A. Lamas-Linares, and C. Kurtsiefer, “Absolute emission rates of spontaneous parametric down-conversion into single transverse Gaussian modes,” Phys. Rev. A 77, 043834 (2008).
[CrossRef]

Langford, N. K.

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]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Law, C. K.

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]

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J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Ling, A.

A. Ling, A. Lamas-Linares, and C. Kurtsiefer, “Absolute emission rates of spontaneous parametric down-conversion into single transverse Gaussian modes,” Phys. Rev. A 77, 043834 (2008).
[CrossRef]

Ma, X.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Mair, A.

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]

Miklaszewski, W.

D. Kaszlikowski, P. Gnacinski, M. Zukowski, W. Miklaszewski, and A. Zeilinger, “Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits,” Phys. Rev. Lett. 85, 4418-4421 (2000).
[CrossRef] [PubMed]

Minardi, S.

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]

Miyamoto, Y.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre Gauss modes using Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[CrossRef] [PubMed]

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Molina-Terriza, G.

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

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]

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[CrossRef]

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]

Monken, C. H.

S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003).
[CrossRef] [PubMed]

C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123-3126 (1998).
[CrossRef]

Nagata, T.

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
[CrossRef] [PubMed]

Nienhuis, G.

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]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Nihira, H.

H. Nihira and C. R. Stroud, Jr., “Robust multipartite multilevel quantum protocols,” Phys. Rev. A 72, 022337 (2005).
[CrossRef]

O'Brien, J. L.

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
[CrossRef] [PubMed]

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Oemrawsingh, S. S.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Oemrawsingh, S. S. R.

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]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

Okamoto, R.

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
[CrossRef] [PubMed]

Osorio, C. I.

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]

Padgett, M.

M. Padgett, J. Courtial, L. Allen, S. Franke-Arnold, S. Barnett, “Entanglement of orbital angular momentum for the signal and idler beams in parametric down-conversion,” J. Mod. Opt. 49, 777-785 (2003).
[CrossRef]

Padgett, M. J.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, “Two-photon entanglement of orbital angular momentum states,” Phys. Rev. A 65, 033823 (2002).
[CrossRef]

J. Leach, M. J. Padgett, S. M. Barnett, S. Franke-Arnold, and J. Courtial, “Measuring the orbital angular momentum of a single photon,” Phys. Rev. Lett. 88, 257901 (2002).
[CrossRef] [PubMed]

Pádua, S.

S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003).
[CrossRef] [PubMed]

C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123-3126 (1998).
[CrossRef]

Pan, J. W.

A. Vaziri, J. W. 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]

Peters, N. A.

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]

Picón, A.

G. F. Calvo, A. Picón, and R. Zambrini, “Measuring the complete transverse spatial mode spectrum of a wave field,” Phys. Rev. Lett. 100, 173902 (2008).
[CrossRef] [PubMed]

G. F. Calvo, A. Picón, and A. Bramon, “Measuring two-photon orbital angular momentum entanglement,” Phys. Rev. A 75, 012319 (2007).
[CrossRef]

Pryde, G. J.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Rehácek, J.

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

Sasaki, K.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre Gauss modes using Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[CrossRef] [PubMed]

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
[CrossRef] [PubMed]

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Shi, X.

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

Souto Ribeiro, P. H.

C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, “Transfer of angular spectrum and image formation in spontaneous parametric down-conversion,” Phys. Rev. A 57, 3123-3126 (1998).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Stroud, C. R.

H. Nihira and C. R. Stroud, Jr., “Robust multipartite multilevel quantum protocols,” Phys. Rev. A 72, 022337 (2005).
[CrossRef]

Stütz, M.

M. Stütz, S. Gröblachera, T. Jennewein, and A. Zeilinger, “How to create and detect N-dimensional entangled photons with an active phase hologram,” Appl. Phys. Lett. 90, 261114 (2007).
[CrossRef]

't Hooft, G. W.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Takeda, M.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre Gauss modes using Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[CrossRef] [PubMed]

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Takeuchi, S.

D. Kawase, Y. Miyamoto, M. Takeda, K. Sasaki, and S. Takeuchi, “Observing quantum correlation of photons in Laguerre Gauss modes using Gouy phase,” Phys. Rev. Lett. 101, 050501 (2008).
[CrossRef] [PubMed]

T. Nagata, R. Okamoto, J. L. O'Brien, K. Sasaki and S. Takeuchi, “Beating the standard quantum limit with four entangled photons,” Science 316, 726 (2007).
[CrossRef] [PubMed]

S. Takeuchi, J. Kim, Y. Yamamoto, and H. H. Hogue, “Development of a high-quantum-efficiency single-photon counting system,” Appl. Phys. Lett. 74, 1063 (1999).
[CrossRef]

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multi-photon counting using visible light photon counter,” Appl. Phys. Lett. 74, 902 (1999).
[CrossRef]

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Torner, L.

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[CrossRef]

J. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003).
[CrossRef]

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]

Torres, J. P.

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

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. P. Torres, A. Alexandrescu, and L. Torner, “Quantum spiral bandwidth of entangled two-photon states,” Phys. Rev. A 68, 050301 (2003).
[CrossRef]

J. P. Torres, Y. Deyanova, L. Torner, and G. Molina-Terriza, “Preparation of engineered two-photon entangled states for multidimensional quantum information,” Phys. Rev. A 67, 052313 (2003).
[CrossRef]

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]

Valencia, A.

A. Valencia, A. Ceré, X. Shi, G. Molina-Terriza, and J. P. Torres, “Shaping the waveform of entangled photons,” Phys. Rev. Lett. 99, 243601 (2007).
[CrossRef]

van Exter, M. P.

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]

Vaziri, A.

G. Molina-Terriza, A. Vaziri, J. Řeháček, Z. Hradil, and A. Zeilinger, “Triggered qutrits for quantum communication protocols,” Phys. Rev. Lett. 92, 167903 (2004).
[CrossRef] [PubMed]

A. Vaziri, J. W. 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]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

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]

Voigt, D.

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

Wada, A.

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Walborn, S. P.

S. P. Walborn, A. N. de Oliveira, S. Pádua, and C. H. Monken, “Multimode Hong-Ou-Mandel interference,” Phys. Rev. Lett. 90, 143601 (2003).
[CrossRef] [PubMed]

Weihs, G.

A. Vaziri, J. W. 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]

A. Vaziri, G. Weihs, and A. Zeilinger, “Experimental two-photon, three-dimensional entanglement for quantum communication,” Phys. Rev. Lett. 89, 240401 (2002).
[CrossRef] [PubMed]

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]

White, A. G.

N. K. Langford, R. B. Dalton, M. D. Harvey, J. L. O'Brien, G. J. Pryde, A. Gilchrist, S. D. Bartlett, and A. G. White, “Measuring entangled qutrits and their use for quantum bit commitment,” Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Woerdman, J. P.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “High-dimensional mode analyzers for spatial quantum entanglement,” Phys. Rev. A 73, 032339 (2006).
[CrossRef]

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]

S. S. Oemrawsingh, X. Ma, D. Voigt, A. Aiello, E. R. Eliel, G. W. 't Hooft, and J. P. Woerdman, “Experimental demonstration of fractional orbital angular momentum entanglement of two photons,” Phys. Rev. Lett. 95, 240501 (2005).
[CrossRef] [PubMed]

S. S. R. Oemrawsingh, A. Aiello, E. R. Eliel, G. Nienhuis, and J. P. Woerdman, “How to observe high-dimensional two-photon entanglement with only two detectors,” Phys. Rev. Lett. 92, 217901 (2004).
[CrossRef] [PubMed]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185-8189 (1992).
[CrossRef] [PubMed]

Yamamoto, Y.

J. Kim, S. Takeuchi, Y. Yamamoto, and H. H. Hogue, “Multi-photon counting using visible light photon counter,” Appl. Phys. Lett. 74, 902 (1999).
[CrossRef]

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Other (6)

These parameters were determined for various experimental conditions.

D. Kawase, S. Takeuchi, K. Sasaki, A. Wada, Y. Miyamoto, and M. Takeda, “Determining the optical axes of entangled Laguerre Gauss modes,” arXiv.org, arXiv:quant-ph/0602199v1.

Note that for ∣m∣>1 there are more than one values of the distance d from the dislocation to the optical axis for which the minimum coincidence probability becomes 0. For the plots in Fig. , we selected the smallest values (d/w0=0.35 for m=2 and 0.26 for m=3) that meet this criterion. We have confirmed that the shape of the plots are almost the same even when we select other values of d for which the minimum coincidence becomes 0.

One can easily confirm this fact by using a simple model assuming the state of the source is given by ∣Ψ⟩=α∣0⟩I∣0⟩S+β∣1⟩I∣−1⟩S and the detection basis states are ∣ψ⟩S=sinδS∣0⟩S+cosδS∣−1⟩S and ∣ψ⟩I=sinδI∣0⟩I+cosδI∣1⟩I.

For the cases where the Rayleigh range of the pump is smaller than the crystal length, L can be considered the Rayleigh range.

This approximation is introduced to simplify the form of the mode function in position representation, which can be easily calculated numerically.

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Figures (5)

Fig. 1
Fig. 1

Experimental setup for measurement of quantum entanglement in spatial modes. The inset shows the scheme of the hologram ( m = 1 ) as viewed from the BBO crystal.

Fig. 2
Fig. 2

2D maps of the coincidence count probabilities P m ( d I , θ I , d S , θ S ) when the position of H S is scanned for four different positions of H I and for three different dislocation orders. The columns show maps for ( d I w 0 , θ I ) = ( 0 , 0 ) , ( 0.5 , 0 ) , ( 1.0 , 0 ) , ( 1.5 , 0 ) , and the rows show maps for m = 1 , 2 , 3 . Parameters are w P w 0 = 2 and b w 0 = 0.102 . The coincidence probabilities are normalized by the coincidence probability for m = 0 in each 2D map.

Fig. 3
Fig. 3

Coincidence fringes when the azimuthal position of H S is scanned, calculated by using Eq. (8) for m = 1 , 2 , 3 with parameters w P w 0 = 2 , b w 0 = 0.102 . The fixed radial positions d S , I w 0 are 0.53 ( m = 1 ) , 0.35 ( m = 2 ) , and 0.26 ( m = 3 ) . The coincidence probability is normalized by the maximum probability in each fringe.

Fig. 4
Fig. 4

Mode distributions in measurement modes in the idler path for m = 1 , 2 , 3 with d I w 0 = 0.53 , 0.35 , 0.26 .

Fig. 5
Fig. 5

(a), (c) Coincidence probabilities when the radial position of H S is scanned as calculated by using Eq. (8) for m = 1 and m = 2 with parameters θ I = θ S = 0 , w P w 0 = 2 , and b w 0 = 0.102 . (b), (d) Coincidence probabilities under the assumption that the crystal length L is zero when the radial position of H S is scanned for m = 1 and m = 2 with parameters θ I = θ S = 0 , w P w 0 = 2

Equations (15)

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Ψ q = d q S d q I Φ ̃ ( q S , q I ) a I ( q S ) a S ( q I ) 0 , 0 ,
Φ ̃ ( q S , q I ) = N q exp ( w P 2 q S + q I 2 4 ) sinc ( L 4 k P q S q I 2 ) ,
Φ ̃ ( q S , q I ) = N q exp ( w P 2 q S + q I 2 4 ) exp ( b 2 q S q I 2 4 ) ,
Φ ( r S , r I ) = d q S d q I Φ ̃ ( q S , q I ) exp ( i q S r S ) exp ( i q I r I ) = N r exp ( ( r S + r I ) 2 2 w P 2 ) exp ( ( r S r I ) 2 2 b 2 ) ,
Ψ r = d r S d r I Φ ( r S , r I ) a I ( r S ) a S ( r I ) 0 , 0 ,
U m ( ρ j , ϕ j ) = exp [ i × sgn ( j ) × m × arg ( ρ j cos ϕ j d j cos θ j , ρ j sin ϕ j d j sin θ j ) ] 2 π 1 w 0 exp ( ρ j 2 w 0 2 ) ,
ψ j m ( d j , θ j ) exp [ i × sgn ( j ) × m × arg ( ρ j cos ϕ j + d j cos θ j , ρ j sin ϕ j + d j sin θ j ) ] 2 π 1 w 0 exp ( ρ j 2 w 0 2 ) a j ( r j ) 0 d r j ,
P m ( d I , θ I , d S , θ S ) = ψ I m ( d I , θ I ) ψ S m ( d S , θ S ) Ψ r 2 d r S d r I Φ ( r S , r I ) exp ( ρ S 2 w 0 2 ) exp ( ρ I 2 w 0 2 ) exp { i × m × [ arg ( ρ I cos ϕ I + d I cos θ I , ρ I sin ϕ I + d I sin θ I ) ] } exp { i × m × [ arg ( ρ S cos ϕ S + d S cos θ S , ρ S sin ϕ S + d I sin θ I ) ] } 2 .
LG p l ( w ; ρ , ϕ ) = 2 p ! π ( l + p ) ! 1 w ( 2 ρ w ) l L p l ( 2 ρ 2 w 2 ) exp ( ρ 2 w 2 + i l ϕ ) ,
P m ( d I , θ I , d S , θ S ) = ψ I m ( d I , θ I ) ψ S m ( d S , θ S ) Ψ r 2 = l I , l S = p I , p S = 0 p I l I ( w ) p S l S ( w ) Ψ r p I l I ( w ) ψ I m ( d I , θ I ) * p S l S ( w ) ψ S m ( d S , θ S ) * 2 .
p l ( w ) = d r LG p l ( w ; r ) a ( r ) 0 ,
p I l I ( w ) p S l S ( w ) Ψ r = δ l S l I C p S l S , p I l I ,
p j l j ( w ) ψ j m ( d j , θ j ) = exp { ( i × sgn ( j ) × m × θ j ) } exp ( i l j θ j ) α p j l j m , j ( d j ) ,
P m = l = exp { i l ( θ I θ S ) } R l m ( d S , d I ) 2 ,
R l m ( d S , d I ) = p S , p I = 0 α p S l m , S * ( d S ) α p I l m , I * ( d I ) C p S l , p I l .

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