Abstract

The orbital angular momentum of photons, being defined in an infinite-dimensional discrete Hilbert space, offers a promising resource for high-dimensional quantum information protocols in quantum optics. The biggest obstacle to its wider use is presently represented by the limited set of tools available for its control and manipulation. Here, we introduce and test experimentally a series of simple optical schemes for the coherent transfer of quantum information from the polarization to the orbital angular momentum of single photons and vice versa. All our schemes exploit a newly developed optical device, the so-called “q-plate”, which enables the manipulation of the photon orbital angular momentum driven by the polarization degree of freedom. By stacking several q-plates in a suitable sequence, one can also have access to higher-order angular momentum subspaces. In particular, we demonstrate the control of the orbital angular momentum m degree of freedom within the subspaces of |m|=2ħ and |m|=4ħ per photon.

© 2009 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (1992).
    [CrossRef] [PubMed]
  2. G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nature Phys. 3, 305-310 (2007).
    [CrossRef]
  3. A. Mair, A. Vaziri, G. Weihs, and A. Zeilinger, "Entanglement of the orbital angular momentum states of photons," Nature(London) 412, 313-316 (2001).
    [CrossRef]
  4. G. Molina-Terriza, J. P. Torres, and L. Torner, "Orbital angular momentum of photons in noncollinear parametric downconversion," Opt. Commun. 228, 155-160 (2003).
    [CrossRef]
  5. 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]
  6. 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, 286-289 (2000)
    [CrossRef] [PubMed]
  7. S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. A 65, 033823 (2002)
    [CrossRef]
  8. M. Stutz, S. Grblacher, 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]
  9. 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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
    [CrossRef] [PubMed]
  10. A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the Orbital Angular Momentum for Applications in Quantum Experiments," Phys. Rev. Lett. 89, 240401 (2002)
    [CrossRef] [PubMed]
  11. A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
    [CrossRef]
  12. 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]
  13. S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. t Hooft, and J. P. Woerdman, "Highdimensional mode analyzers for spatial quantum entanglement," Phys. Rev. A 73, 032339 (2006).
    [CrossRef]
  14. 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]
  15. J. T. Barreiro, T. C. Wei, and P. G. Kwiat, "Beating the channel capacity limit for linear photonic superdense coding," Nature Phys. 4, 282-286 (2008).
    [CrossRef]
  16. L. Chen and W. She, "Increasing Shannon dimensionality by hyperentanglement of spin and fractional orbital angular momentum," Opt. Lett. 34, 1855-1857 (2009).
    [CrossRef] [PubMed]
  17. L. Marrucci, C. Manzo, D. and Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  18. E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
    [CrossRef] [PubMed]
  19. J. B. Gotte, K. OHolleran, D. Preece, F. Flossmann, S. Franke-Arnold, S. M. Barnett, and M. J. Padgett, "Light beams with fractional orbital angular momentum and their vortex structure," Opt. Express 16, 993-1006 (2008)
    [CrossRef] [PubMed]
  20. L. Marrucci, C. Manzo, and D. Paparo, "Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation," Appl. Phys. Lett. 88, 221102 (2006).
    [CrossRef]
  21. E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
    [CrossRef]
  22. G. F. Calvo, and A. Picon, "Spin-induced angular momentum switching," Opt. Lett. 32, 838-840 (2007)
    [CrossRef] [PubMed]
  23. E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, "Light propagation in a birefringent plate with topological charge," Opt. Lett. 34, 1225-1227 (2009).
    [CrossRef] [PubMed]
  24. E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian Modes," Opt. Lett. 32, 3053-3055 (2007).
    [CrossRef] [PubMed]
  25. P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
    [CrossRef] [PubMed]
  26. M. J. Padgett, and J. Courtial, "New high-intensity source of polarization-entangled photon pairs," Opt. Lett. 24, 430 (1999).
    [CrossRef]
  27. We note that, although the optical layout is a Mach-Zehnder interferometer, the optical path phase difference between the two arms of the interferometer is only affecting the polarization state of the single output obtained after the final PBS, while it does not act on the PBS exit mode and on the OAM final state. The final polarization may therefore turn elliptical if this phase difference is not well controlled. However, the H polarization can be easily restored by suitable wave-plates, as long as it is uniform.
  28. M. Fiorentino, and F. N. C. Wong, "Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic," Phys. Rev. Lett. 93, 070502 (2004).
    [CrossRef] [PubMed]

2009 (4)

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, "Light propagation in a birefringent plate with topological charge," Opt. Lett. 34, 1225-1227 (2009).
[CrossRef] [PubMed]

L. Chen and W. She, "Increasing Shannon dimensionality by hyperentanglement of spin and fractional orbital angular momentum," Opt. Lett. 34, 1855-1857 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (4)

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

M. Stutz, S. Grblacher, 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]

G. F. Calvo, and A. Picon, "Spin-induced angular momentum switching," Opt. Lett. 32, 838-840 (2007)
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian Modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

2006 (3)

L. Marrucci, C. Manzo, and D. Paparo, "Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation," Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

L. Marrucci, C. Manzo, D. and Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. t Hooft, and J. P. Woerdman, "Highdimensional mode analyzers for spatial quantum entanglement," Phys. Rev. A 73, 032339 (2006).
[CrossRef]

2005 (3)

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]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
[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]

2004 (2)

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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

M. Fiorentino, and F. N. C. Wong, "Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic," Phys. Rev. Lett. 93, 070502 (2004).
[CrossRef] [PubMed]

2003 (2)

G. Molina-Terriza, J. P. Torres, and L. Torner, "Orbital angular momentum of photons in noncollinear parametric downconversion," Opt. Commun. 228, 155-160 (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]

2002 (2)

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the Orbital Angular Momentum for Applications in Quantum Experiments," Phys. Rev. Lett. 89, 240401 (2002)
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. A 65, 033823 (2002)
[CrossRef]

2001 (1)

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

2000 (1)

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, 286-289 (2000)
[CrossRef] [PubMed]

1999 (1)

1995 (1)

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

1992 (1)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (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, "Highdimensional mode analyzers for spatial quantum entanglement," Phys. Rev. A 73, 032339 (2006).
[CrossRef]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
[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]

Allen, L.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. A 65, 033823 (2002)
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Arnaut, 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, 286-289 (2000)
[CrossRef] [PubMed]

Barbosa, G. A.

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, 286-289 (2000)
[CrossRef] [PubMed]

Barnett, S. M.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. A 65, 033823 (2002)
[CrossRef]

Barreiro, J. T.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, "Beating the channel capacity limit for linear photonic superdense coding," Nature Phys. 4, 282-286 (2008).
[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]

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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Calvo, G. F.

Chen, L.

Courtial, J.

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, "Observation of quantum entanglement using spatial light modulators," 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, "Highdimensional mode analyzers for spatial quantum entanglement," Phys. Rev. A 73, 032339 (2006).
[CrossRef]

De Martini, F.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[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, "Highdimensional mode analyzers for spatial quantum entanglement," Phys. Rev. A 73, 032339 (2006).
[CrossRef]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
[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]

Fiorentino, M.

M. Fiorentino, and F. N. C. Wong, "Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic," Phys. Rev. Lett. 93, 070502 (2004).
[CrossRef] [PubMed]

Franke-Arnold, S.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," 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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Gotte, J. B.

Grblacher, S.

M. Stutz, S. Grblacher, 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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Jennewein, T.

M. Stutz, S. Grblacher, 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]

Karimi, E.

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, "Light propagation in a birefringent plate with topological charge," Opt. Lett. 34, 1225-1227 (2009).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian Modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

Kwiat, P. G.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, "Beating the channel capacity limit for linear photonic superdense coding," Nature Phys. 4, 282-286 (2008).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

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]

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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Ma, X.

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. t Hooft, and J. P. Woerdman, "Highdimensional 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(London) 412, 313-316 (2001).
[CrossRef]

Manzo, C.

L. Marrucci, C. Manzo, and D. Paparo, "Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation," Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

L. Marrucci, C. Manzo, D. and Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

Marrucci, L.

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, "Light propagation in a birefringent plate with topological charge," Opt. Lett. 34, 1225-1227 (2009).
[CrossRef] [PubMed]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian Modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, D. and Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

L. Marrucci, C. Manzo, and D. Paparo, "Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation," Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

Mattle, K.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

Molina-Terriza, G.

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

G. Molina-Terriza, J. P. Torres, and L. Torner, "Orbital angular momentum of photons in noncollinear parametric downconversion," Opt. Commun. 228, 155-160 (2003).
[CrossRef]

Nagali, E.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

O’Brien, J. L.

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, "Observation of quantum entanglement using spatial light modulators," 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, "Highdimensional 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.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
[CrossRef]

Padgett, M. J.

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. A 65, 033823 (2002)
[CrossRef]

M. J. Padgett, and J. Courtial, "New high-intensity source of polarization-entangled photon pairs," Opt. Lett. 24, 430 (1999).
[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]

Paparo, D.

L. Marrucci, C. Manzo, and D. Paparo, "Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation," Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

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]

Piccirillo, B.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, "Light propagation in a birefringent plate with topological charge," Opt. Lett. 34, 1225-1227 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian Modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

Picon, A.

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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Santamato, E.

E. Karimi, B. Piccirillo, L. Marrucci, and E. Santamato, "Light propagation in a birefringent plate with topological charge," Opt. Lett. 34, 1225-1227 (2009).
[CrossRef] [PubMed]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

E. Karimi, G. Zito, B. Piccirillo, L. Marrucci, and E. Santamato, "Hypergeometric-Gaussian Modes," Opt. Lett. 32, 3053-3055 (2007).
[CrossRef] [PubMed]

Sciarrino, F.

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

Sergienko, A. V.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

She, W.

Shih, Y. H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Stutz, M.

M. Stutz, S. Grblacher, 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]

Torner, L.

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

G. Molina-Terriza, J. P. Torres, and L. Torner, "Orbital angular momentum of photons in noncollinear parametric downconversion," Opt. Commun. 228, 155-160 (2003).
[CrossRef]

Torres, J. P.

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

G. Molina-Terriza, J. P. Torres, and L. Torner, "Orbital angular momentum of photons in noncollinear parametric downconversion," Opt. Commun. 228, 155-160 (2003).
[CrossRef]

Vaziri, A.

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, "Superpositions of the Orbital Angular Momentum for Applications in Quantum Experiments," 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(London) 412, 313-316 (2001).
[CrossRef]

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]

Wei, T. C.

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, "Beating the channel capacity limit for linear photonic superdense coding," Nature Phys. 4, 282-286 (2008).
[CrossRef]

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, "Superpositions of the Orbital Angular Momentum for Applications in Quantum Experiments," 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(London) 412, 313-316 (2001).
[CrossRef]

Weinfurter, H.

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

Woederman, J. P.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

Woerdman, J. P.

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
[CrossRef]

Wong, F. N. C.

M. Fiorentino, and F. N. C. Wong, "Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic," Phys. Rev. Lett. 93, 070502 (2004).
[CrossRef] [PubMed]

Zeilinger, A.

M. Stutz, S. Grblacher, 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]

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the Orbital Angular Momentum for Applications in Quantum Experiments," 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(London) 412, 313-316 (2001).
[CrossRef]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[CrossRef] [PubMed]

Zito, G.

Appl. Phys. Lett. (3)

M. Stutz, S. Grblacher, 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]

L. Marrucci, C. Manzo, and D. Paparo, "Pancharatnam-Berry phase optical elements for wavefront shaping in the visible domain: switchable helical modes generation," Appl. Phys. Lett. 88, 221102 (2006).
[CrossRef]

E. Karimi, B. Piccirillo, E. Nagali, L. Marrucci, and E. Santamato, "Efficient generation and sorting of orbital angular momentum eigenmodes of light by thermally tuned q-plates," Appl. Phys. Lett. 94, 231124 (2009).
[CrossRef]

Nature Phys. (2)

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

J. T. Barreiro, T. C. Wei, and P. G. Kwiat, "Beating the channel capacity limit for linear photonic superdense coding," Nature Phys. 4, 282-286 (2008).
[CrossRef]

Nature(London) (1)

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

Opt. Commun. (1)

G. Molina-Terriza, J. P. Torres, and L. Torner, "Orbital angular momentum of photons in noncollinear parametric downconversion," Opt. Commun. 228, 155-160 (2003).
[CrossRef]

Opt. Express (1)

Opt. Lett. (5)

Phys. Rev. A (4)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woederman, "Spin-orbit coupling in free-space Laguerre-Gaussian light beams," Phys. Rev. A 45, 8185 (1992).
[CrossRef] [PubMed]

S. Franke-Arnold, S. M. Barnett, M. J. Padgett, and L. Allen, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. A 65, 033823 (2002)
[CrossRef]

A. Aiello, S. S. R. Oemrawsingh, E. R. Eliel, and J. P. Woerdman, "Nonlocality of high-dimensional two-photon orbital angular momentum states," Phys. Rev. A 72, 052114 (2005).
[CrossRef]

S. S. Oemrawsingh, J. A. de Jong, X. Ma, A. Aiello, E. R. Eliel, G. W. t Hooft, and J. P. Woerdman, "Highdimensional mode analyzers for spatial quantum entanglement," Phys. Rev. A 73, 032339 (2006).
[CrossRef]

Phys. Rev. Lett. (10)

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]

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]

L. Marrucci, C. Manzo, D. and Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
[CrossRef] [PubMed]

E. Nagali, F. Sciarrino, F. De Martini, L. Marrucci, B. Piccirillo, E. Karimi, and E. Santamato, "Quantum information transfer from spin to orbital angular momentum of photons," Phys. Rev. Lett. 103, 013601 (2009).
[CrossRef] [PubMed]

P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and Y. H. Shih, "New high-intensity source of polarization-entangled photon pairs," Phys. Rev. Lett. 75, 4337 (1995).
[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, "Observation of quantum entanglement using spatial light modulators," Phys. Rev. Lett. 93, 053601 (2004).
[CrossRef] [PubMed]

A. Vaziri, G. Weihs, and A. Zeilinger, "Superpositions of the Orbital Angular Momentum for Applications in Quantum Experiments," Phys. Rev. Lett. 89, 240401 (2002)
[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]

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, 286-289 (2000)
[CrossRef] [PubMed]

M. Fiorentino, and F. N. C. Wong, "Deterministic Controlled-NOT Gate For Single-Photon Two-Qubit Quantum Logic," Phys. Rev. Lett. 93, 070502 (2004).
[CrossRef] [PubMed]

Other (1)

We note that, although the optical layout is a Mach-Zehnder interferometer, the optical path phase difference between the two arms of the interferometer is only affecting the polarization state of the single output obtained after the final PBS, while it does not act on the PBS exit mode and on the OAM final state. The final polarization may therefore turn elliptical if this phase difference is not well controlled. However, the H polarization can be easily restored by suitable wave-plates, as long as it is uniform.

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Figures (6)

Fig. 1.
Fig. 1.

Schematic representation of the experimental setup. Outside the dashed box is the first section of the apparatus, common to all our experiments. The main optical source is a ultraviolet (UV) beam with wavelength λp =397.5 nm, which is used as pump beam for the photon pairs generation. The UV beam pumps a 1.5 mm thick nonlinear crystal of β -barium borate (BBO) cut for type II phase-matching, working in a collinear regime and generating polarization pairs of photons with the same wavelength λ and orthogonal linear polarizations, hereafter denoted as horizontal (H) and vertical (V). The spatial and temporal walk-offs are compensated by a half-wave plate and a 0.75 mm thick BBO (CW ) [25]. Finally, the photons are spectrally filtered by an interference filter with bandwidth Δλ=6 nm. In order to work in the one-photon regime, a polarizing beam-splitter (PBS) transmits the horizontally-polarized photon of the pair and reflects the vertically-polarized one. The latter is then coupled to a single-mode fiber and revealed with a single-photon counter module (SPCM), which therefore acts as a trigger of the one-photon state generation. In the dashed box, the four configurations a, b, c, d of the second section of the apparatus are shown, used in the four experiments discussed in this paper. Legend: C - fiber compensation stage; DT trigger detection unit; QP - q-plate; Hol - hologram; P.A. - polarization analysis set, as shown in solid-line box.

Fig. 2.
Fig. 2.

Patterns of the 12 binary holograms used in this work. The left box refers to the OAM subspace o 2 (|m|=2). The right box to the OAM subspace o 4 (|m|=4). In the upper-left corner of each hologram is shown the quantum state that is generated by the hologram, when using a TEM00 input, in the first-order diffraction beam.

Fig. 3.
Fig. 3.

Experimental density matrices ρ (the left column shows the real part and right column the imaginary part) measured for the output of the πo 2 qubit transfer, for each of the three different predicted output states shown in the upper left corner of each row.

Fig. 4.
Fig. 4.

Left Side -Experimental density matrices ρ (the left column shows the real part and right column the imaginary part) measured for the output of the o 2π qubit transfer, for each of the three different predicted output states shown in the upper left corner of each row. Right Side -Experimental density matrices measured in the polarization degree of freedom after the bidirectional πo 2π transferrer. In each box is reported the expression of the initial and final state, to be compared with the experimental one described by the density matrix.

Fig. 5.
Fig. 5.

a)Schematic representation of deterministic transferrer: A pair of suitably rotated Dove’s prisms (DP), combined with wave plates for polarization compensation, are used to realize a sigmaz operation on the OAM degree of freedom. The transferrer converts the polarization in OAM if the light goes from left to right, while it converts the OAM into polarization if crossed from right to left. b)Alternative experimental scheme for the deterministic transferrer based on a Sagnac interferometer.

Fig. 6.
Fig. 6.

Experimental density matrices measured in the OAM basis {|+4〉,|-4〉} for different predicted final states, shown in the lower-left corner of each panel.

Tables (4)

Tables Icon

Table 1. Fidelity values between the experimental states generated by the πo 2 transferrer and the theoretical ones expected after the conversion in the OAMdegree of freedom of the qubit initially encoded in the polarization.

Tables Icon

Table 2. Fidelity values between the experimental states generated by the o 2π transferrer and the theoretical ones expected after the conversion in polarization degree of freedom of the qubit initially encoded in the OAM.

Tables Icon

Table 3. Fidelity values between the input and output states for the bidirectional πo 2π transferrer.

Tables Icon

Table 4. Fidelity values between the expected and the experimental states generated by the πo 4 transferrer.

Equations (28)

Equations on this page are rendered with MathJax. Learn more.

|L π|moQp|Rπ|m+2o
|R π |m o QP |L π |m 2 o
α|L π |m o + β |R π |m oQPα|Rπ|m+2o+β|Lπ|m2o
ψ0(r,φ,s)=usf(r)QPψ1(r,φ,s)=eimφusf(r)
ψ1(r,φ,s)=eimφusf(r)airpropagationacrosszeimφusF|m|(r,z)
F|m|(r,z)=0A|m|(k)J|m|(kr)eizk02k2kdk,
A|m|(k)=0f(r)J|m|(kr)kdk.
(α|m+β|m)usf(r)airpropagationacrossz(α|m+β|m)usF|m|(r,z)
|h=12(|l+|r)
|υ=1i2(|l|r)
|a=12(|h+|υ)=eiπ42(|l+i|r)
|d=12(|h|υ)=eiπ42(|li|r).
Ψin=φπ0o=(αHπ+βVπ)0o
(αLπ+βRπ)0o
αR+2+β L 2.
Ψout=Hπ(α+2o+β2o)=Hπφo2,
Ψin=Hπφo2=H (α+2+β2)
12{αV+4+αH0+βV0+βH4}
Ψout=(αH+βV)0=φπ0o
Ψin=φπ0o=(αH+βV)0
αR+2+βL2
12 (H(α+2+β2)+V(α+2β2))
H12(α+2+β2)+V12(α+V12(α+2+β2))
A(α+2+β2)
Ψout=H(α+2+β2)=Hπφo2
φπ0l(αR2+βL+2)
αL+2+βR2
(α+4+β4)H=φo4Hπ

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