Abstract

We consider the possibility of performing quantum key distribution (QKD) by encoding information onto individual photons using plane-wave basis states. We compare the results of this calculation to those obtained by earlier workers, who considered encoding using OAM-carrying vortex modes of the field. We find theoretically that plane-wave encoding is less strongly influenced by atmospheric turbulence than is OAM encoding, with potentially important implications for free-space quantum key distribution.

© 2011 OSA

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  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]
  2. N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
    [CrossRef]
  3. 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]
  4. M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (2002).
    [CrossRef]
  5. 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]
  6. 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]
  7. M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
    [CrossRef]
  8. J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
    [CrossRef]
  9. J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
    [CrossRef] [PubMed]
  10. B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
    [CrossRef]
  11. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
    [CrossRef] [PubMed]
  12. C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
    [CrossRef]
  13. G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–260 (2008).
    [CrossRef]
  14. G. A. Tyler and R. W. Boyd, “Influence of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009). To allow straightforward comparison between the OAM and plane-wave cases, in the present article we use notation similar to that of this earlier paper.
    [CrossRef] [PubMed]
  15. C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
    [CrossRef]
  16. B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011).
    [CrossRef] [PubMed]
  17. F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
    [CrossRef]
  18. Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
    [CrossRef]
  19. Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
    [CrossRef]
  20. We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis.
    [CrossRef] [PubMed]
  21. W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. 11, 0313–0325 (2011).
  22. D. L. Fried, “Optical resolution through a randomly inhomogeneous medium for very short and very long exposures,” J. Opt. Soc. Am. 56, 1372–1379 (1966).
    [CrossRef]
  23. The propagation of light through atmospheric turbulence is reviewed by P. W. Milonni, “Adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
    [CrossRef]

2011

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[CrossRef]

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
[CrossRef]

Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
[CrossRef]

W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. 11, 0313–0325 (2011).

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011).
[CrossRef] [PubMed]

2010

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

2009

2008

We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis.
[CrossRef] [PubMed]

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
[CrossRef]

G. Gbur and R. K. Tyson, “Vortex beam propagation through atmospheric turbulence and topological charge conservation,” J. Opt. Soc. Am. A 25, 225–260 (2008).
[CrossRef]

2007

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[CrossRef]

2006

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

2005

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

2002

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]

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]

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (2002).
[CrossRef]

2001

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]

1999

The propagation of light through atmospheric turbulence is reviewed by P. W. Milonni, “Adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
[CrossRef]

1992

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]

1966

Allen, L.

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]

Andrews, R.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

Barnett, S. M.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[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]

Bjork, G.

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (2002).
[CrossRef]

Bonato, C.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Bourennane, M.

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (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]

Boyd, R. W.

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]

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (2002).
[CrossRef]

Courtial, J.

J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Da Deppo, V.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Denis, M. R.

J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Dymale, R. C.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
[CrossRef]

Eliel, E. R.

Franke-Arnold, S.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

Fried, D. L.

Gbur, G.

Gisin, N.

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[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]

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (2002).
[CrossRef]

Gopaul, C.

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

Gruneisen, M. T.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
[CrossRef]

Ireland, D. G.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

Jack, B.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

Jha, A. K.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

Jia, J.

Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
[CrossRef]

Jia, J.-J

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
[CrossRef]

Karlsson, A.

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (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]

Leach, J.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

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]

Miller, W. A.

W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. 11, 0313–0325 (2011).

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
[CrossRef]

Milonni, P. W.

The propagation of light through atmospheric turbulence is reviewed by P. W. Milonni, “Adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
[CrossRef]

Molina-Terriza, G.

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.

Naletto, G.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Padgett, M. J.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Paterson, C.

C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
[CrossRef] [PubMed]

Pors, B.-J.

Raymer, M. G.

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[CrossRef]

Romero, J.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

Roux, F. S.

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
[CrossRef]

Silberhorn, C.

We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis.
[CrossRef] [PubMed]

Smith, B. J.

B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
[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]

Sweiti, A. M.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
[CrossRef]

Thew, R.

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[CrossRef]

Tomaello, A.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Torner, L.

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.

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]

Tyler, G. A.

Tyson, R. K.

Vaziri, 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]

Villoresi, P.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

Walmsley, I. A.

We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis.
[CrossRef] [PubMed]

Wang, J.

Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
[CrossRef]

Wang, J.-Y.

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
[CrossRef]

Wang, Y.

Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
[CrossRef]

Wang, Y.-G.

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
[CrossRef]

Weihs, G.

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]

Woerdman, J. P.

B.-J. Pors, C. H. Monken, E. R. Eliel, and J. P. Woerdman, “Transport of orbital-angular-momentum entanglement through a turbulent atmosphere,” Opt. Express 19, 6671–6683 (2011).
[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]

Xu, J.-C.

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
[CrossRef]

Yao, A. M.

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

Zeilinger, 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]

Zhang, L.

We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis.
[CrossRef] [PubMed]

Zhang, Y.

Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
[CrossRef]

Zhang, Y.-X.

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
[CrossRef]

Am. J. Phys.

The propagation of light through atmospheric turbulence is reviewed by P. W. Milonni, “Adaptive optics for astronomy,” Am. J. Phys. 67, 476–485 (1999).
[CrossRef]

Appl. Opt.

M. T. Gruneisen, W. A. Miller, R. C. Dymale, and A. M. Sweiti, “Holographic generation of complex fields with spatial light modulators: application to quantum key distribution,” Appl. Opt. 47, A33–A41 (2008).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys. A

M. Bourennane, A. Karlsson, G. Bjork, N. Gisin, and N. J. Cerf, “Quantum key distribution using multilevel encoding: security analysis,” J. Phys. A 35, 10065–10076 (2002).
[CrossRef]

Nat. Photonics

N. Gisin and R. Thew, “Quantum communication,” Nat. Photonics 1, 165–171 (2007).
[CrossRef]

Nature

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]

New J. Phys.

C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11, 045017 (2009).
[CrossRef]

C. Gopaul and R. Andrews, “The effect of atmospheric turbulence on entangled orbital angular momentum states,” New J. Phys. 9, 94 (2007).
[CrossRef]

J. Leach, M. R. Denis, J. Courtial, and M. J. Padgett, “Vortex knots in light,” New J. Phys. 7, 55 (2005).
[CrossRef]

Opt. Commun.

Y.-X. Zhang, Y.-G. Wang, J.-C. Xu, J.-Y. Wang, and J.-J Jia, “Orbital angular momentum crosstalk of single photons propagation in a slant non-Kolmogorov turbulence channel,” Opt. Commun. 284, 1132–1138 (2011).
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Opt. Express

Opt. Laser Technol.

Y. Wang, Y. Zhang, J. Wang, and J. Jia, “Degree of polarization for quantum light field propagating through non-Kolmogorov turbulence,” Opt. Laser Technol. 43, 776–780 (2011).
[CrossRef]

Opt. Lett.

Phys. Rev. A

F. S. Roux, “Infinitesimal-propagation equation for decoherence of an orbital-angular-momentum-entangled biphoton state in atmospheric turbulence,” Phys. Rev. A 83, 053822 (2011).
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B. J. Smith and M. G. Raymer, “Two-photon wave mechanics,” Phys. Rev. A 74, 062104 (2006).
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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).
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Phys. Rev. Lett.

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).
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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).
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C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005).
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We note that a security analysis of such a plane-wave encoding scheme has been presented earlier for the case of a continuous-variable protocol by L. Zhang, C. Silberhorn, and I. A. Walmsley, “Secure quantum key distribution using continuous variables of single photons,” Phys. Rev. Lett. 100, 110504 (2008). In the present paper we discretize the propagation direction to better compare our results to those of earlier work based on an OAM basis.
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Quantum Inf. Comput.

W. A. Miller, “Efficient photon sorter in a high-dimensional state space,” Quantum Inf. Comput. 11, 0313–0325 (2011).

Science

J. Leach, B. Jack, J. Romero, A. K. Jha, A. M. Yao, S. Franke-Arnold, D. G. Ireland, R. W. Boyd, S. M. Barnett, and M. J. Padgett, “Quantum correlations in optical angle-orbital angular momentum variables,” Science 329, 662–665 (2010).
[CrossRef] [PubMed]

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Figures (2)

Fig. 1
Fig. 1

Schematic diagram of a free-space quantum communication link.

Fig. 2
Fig. 2

The quantity 〈s Δ〉 plotted against the strength of the atmospheric turbulence as quantified by the ratio of the linear size L of the telescope aperture to the Fried parameter r 0 for several values of Δ. 〈s Δ〉 is the ensemble average of the fraction of the received power that is found to be in plane-wave mode n = + Δ, assuming that the transmitted beam was in plane-wave mode . Solid lines give the predictions based on a numerical evaluation of the integral in Eq. (24). The dashed curve refers to the OAM case treated in reference [14]. The quantity L represents the diameter of the circular aperture for the OAM case and the length of each side of the square aperture for the plane wave case. Note that the the plane wave encoding is more robust (by about as much as a factor of three) than the OAM encoding.

Equations (28)

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E n = A exp ( i k z + i m q x ) .
O m n = L / 2 L / 2 E m E n * d x ,
O m n = | A | 2 L / 2 L / 2 e i q ( m n ) x d x .
O m n = | A | 2 2 sin [ q ( m n ) L / 2 ] q ( m n ) .
O m n = | A | 2 sin [ π ( m n ) ] π ( m n ) / L ,
A ( x , y ) = A 0 W ( x / L ) W ( y / L ) e i l q x ,
V ( x , y ) = A 0 W ( x / L ) W ( y / L ) e i l q x e i ϕ ( x , y ) ,
e i ϕ ( x , y ) = m = g m ( y ) e i m q x ,
g m ( y ) = 1 L L / 2 L / 2 d x e i ϕ ( x , y ) e i m q x .
V ( x , y ) = n = V n ( y ) e i n q x ,
V n ( y ) = 1 L L / 2 L / 2 d x V ( x , y ) e i n q x .
V n ( y ) = A 0 W ( y / L ) m = g m ( y ) L / 2 L / 2 d x e i ( l + m n ) q x .
V n ( y ) = A 0 W ( y / L ) g n l ( y ) .
P = 1 2 ε 0 c d x d y V * ( r ) V ( r ) = 1 2 ε 0 c | A 0 | 2 L 2 .
P = Δ = P Δ
P Δ = 1 2 ε 0 c | A 0 | 2 L L / 2 L / 2 d y g Δ * ( y ) g Δ ( y )
s Δ = 1 L L / 2 L / 2 d y g Δ * ( y ) g Δ ( y ) .
s Δ = 1 L L / 2 L / 2 d y g Δ * ( y ) g Δ ( y )
s Δ = 1 L L / 2 L / 2 d y 1 L L / 2 L / 2 d x 1 e i ϕ ( x 1 , y ) e i Δ q x 1 1 L L / 2 L / 2 d x 2 e i ϕ ( x 2 , y ) e i Δ q x 2 = 1 L 3 L / 2 L / 2 d y L / 2 L / 2 d x 1 L / 2 L / 2 d x 2 e i ϕ ( x 1 , y ) e i ϕ ( x 2 , y ) e i Δ q ( x 1 x 2 )
e i [ ϕ ( x 1 , y ) ϕ ( x 2 , y ) ] = e 1 2 [ ϕ ( x 1 , y ) ϕ ( x 2 , y ) ] 2 .
[ ϕ ( x 1 , y ) ϕ ( x 2 , y ) ] 2 = 6.88 | x 1 x 2 r 0 | 5 / 3 ,
s Δ = 1 L 3 d y d x 1 d x 2 e i Δ q ( x 1 x 2 ) e 3.44 ( | x 2 x 1 | / r 0 ) 5 / 3
s Δ = 2 1 2 1 2 d η ( 1 2 + | η | 1 2 | η | d ζ ) e 3.44 ( L / r 0 ) 5 / 3 | η | 5 / 3 e 2 i Δ L q η
s Δ = 8 0 1 2 d η ( 1 2 η ) e 3.44 ( η L / r 0 ) 5 / 3 cos ( 4 π Δ η ) .
s 0 = 1 0.22 ( L / r 0 ) 5 / 3 .
s Δ = 0.0674 Δ 2 ( L r 0 ) 5 / 3 .
s Δ = 1.70 ( r 0 / L )
s 0 = [ 1 + ( 0.659 L / r 0 ) 2 ] 1 / 2 .

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