Abstract

The topological phase acquired by vector vortex optical beams is investigated. Under local unitary operations on their polarization and transverse degrees of freedom, the vector vortices can only acquire discrete geometric phase values, 0 or π, associated with closed paths belonging to different homotopy classes on the SO(3) manifold. These discrete values are demonstrated through interferometric measurements, and the spin-orbit mode separability is associated to the visibility of the interference patterns. The local unitary operations performed on the vector vortices involved both polarization and transverse mode transformations with birefringent wave plates and astigmatic mode converters. The experimental results agree with our theoretical simulations and generalize our previous results obtained with polarization transformations only.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).
  2. S. Pancharatnam, Collected Works of S. Pancharatnam (Oxford University, 1975).
  3. Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
    [CrossRef]
  4. M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
    [CrossRef]
  5. J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
    [CrossRef]
  6. L.-M. Duan, J. I. Cirac, and P. Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
    [CrossRef]
  7. J.-Q. Zhang, Y.-F. Yu, and Z.-M. Zhang, “Unconventional geometric phase gate with two nonidentical quantum dots trapped in a photonic crystal cavity,” J. Opt. Soc. Am. B 28, 1959–1963 (2011).
    [CrossRef]
  8. S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
    [CrossRef]
  9. M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
    [CrossRef]
  10. E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
    [CrossRef]
  11. P. Kurzynowski, W. A. Woźniak, and M. Szarycz, “Geometric phase: two triangles on the Poincar sphere,” J. Opt. Soc. Am. A 28, 475–482 (2011).
    [CrossRef]
  12. E. Sjöqvist, “Geometric phase for entangled spin pairs,” Phys. Rev. A 62, 022109 (2000).
    [CrossRef]
  13. B. Hessmo and E. Sjöqvist, “Quantal phase for nonmaximally entangled photons,” Phys. Rev. A 62, 062301 (2000).
    [CrossRef]
  14. D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
    [CrossRef]
  15. E. Sjöqvist, “Entanglement-induced geometric phase of quantum states,” Phys. Lett. A 374, 1431–1433 (2010).
    [CrossRef]
  16. S. A. Schulz, T. Machula, E. Karimi, and R. W. Boyd, “Integrated multi vector vortex beam generator,” Opt. Express 21, 16130–16141 (2013).
    [CrossRef]
  17. F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
    [CrossRef]
  18. F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
    [CrossRef]
  19. L. Aolita and S. P. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
    [CrossRef]
  20. C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
    [CrossRef]
  21. V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
    [CrossRef]
  22. G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
    [CrossRef]
  23. G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
    [CrossRef]
  24. P. Milman and R. Mosseri, “Topological phase for entangled two-qubit states,” Phys. Rev. Lett. 90, 230403 (2003).
    [CrossRef]
  25. P. Milman, “Phase dynamics of entangled qubits,” Phys. Rev. A 73, 062118 (2006).
    [CrossRef]
  26. R. Mosseri and R. Dandoloff, “Geometry of entangled states, Bloch spheres and Hopf fibrations,” J. Phys. A 34, 10243–10252 (2001).
    [CrossRef]
  27. C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
    [CrossRef]
  28. J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
    [CrossRef]
  29. L. E. Oxman and A. Z. Khoury, “Fractional topological phase for entangled qudits,” Phys. Rev. Lett. 106, 240503 (2011).
    [CrossRef]
  30. M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
    [CrossRef]
  31. A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
    [CrossRef]
  32. M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
    [CrossRef]
  33. A. E. Siegman, Lasers (University Science Books, 1986).
  34. A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).
  35. C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
    [CrossRef]
  36. R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
    [CrossRef]
  37. R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
    [CrossRef]
  38. C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
    [CrossRef]
  39. L. Chen and W. She, “Single-photon spin-orbit entanglement violating a Bell-like inequality,” J. Opt. Soc. Am. B 27, A7–A10 (2010).
    [CrossRef]
  40. E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
    [CrossRef]
  41. K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
    [CrossRef]
  42. E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
    [CrossRef]
  43. 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]
  44. M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  45. C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express 18, 9207–9212 (2010).
    [CrossRef]
  46. W. LiMing, Z. L. Tang, and C. J. Liao, “Representation of the SO(3) group by a maximally entangled state,” Phys. Rev. A 69, 064301 (2004).
    [CrossRef]
  47. N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
    [CrossRef]
  48. N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
    [CrossRef]

2013 (4)

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
[CrossRef]

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[CrossRef]

S. A. Schulz, T. Machula, E. Karimi, and R. W. Boyd, “Integrated multi vector vortex beam generator,” Opt. Express 21, 16130–16141 (2013).
[CrossRef]

2012 (4)

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[CrossRef]

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[CrossRef]

2011 (5)

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

L. E. Oxman and A. Z. Khoury, “Fractional topological phase for entangled qudits,” Phys. Rev. Lett. 106, 240503 (2011).
[CrossRef]

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

P. Kurzynowski, W. A. Woźniak, and M. Szarycz, “Geometric phase: two triangles on the Poincar sphere,” J. Opt. Soc. Am. A 28, 475–482 (2011).
[CrossRef]

J.-Q. Zhang, Y.-F. Yu, and Z.-M. Zhang, “Unconventional geometric phase gate with two nonidentical quantum dots trapped in a photonic crystal cavity,” J. Opt. Soc. Am. B 28, 1959–1963 (2011).
[CrossRef]

2010 (5)

L. Chen and W. She, “Single-photon spin-orbit entanglement violating a Bell-like inequality,” J. Opt. Soc. Am. B 27, A7–A10 (2010).
[CrossRef]

C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express 18, 9207–9212 (2010).
[CrossRef]

E. Sjöqvist, “Entanglement-induced geometric phase of quantum states,” Phys. Lett. A 374, 1431–1433 (2010).
[CrossRef]

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[CrossRef]

2008 (1)

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

2007 (3)

L. Aolita and S. P. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
[CrossRef]

C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[CrossRef]

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

2006 (1)

P. Milman, “Phase dynamics of entangled qubits,” Phys. Rev. A 73, 062118 (2006).
[CrossRef]

2004 (1)

W. LiMing, Z. L. Tang, and C. J. Liao, “Representation of the SO(3) group by a maximally entangled state,” Phys. Rev. A 69, 064301 (2004).
[CrossRef]

2003 (3)

P. Milman and R. Mosseri, “Topological phase for entangled two-qubit states,” Phys. Rev. Lett. 90, 230403 (2003).
[CrossRef]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

2001 (4)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[CrossRef]

L.-M. Duan, J. I. Cirac, and P. Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
[CrossRef]

R. Mosseri and R. Dandoloff, “Geometry of entangled states, Bloch spheres and Hopf fibrations,” J. Phys. A 34, 10243–10252 (2001).
[CrossRef]

F. Gori, “Polarization basis for vortex beams,” J. Opt. Soc. Am. A 18, 1612–1617 (2001).
[CrossRef]

2000 (3)

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
[CrossRef]

E. Sjöqvist, “Geometric phase for entangled spin pairs,” Phys. Rev. A 62, 022109 (2000).
[CrossRef]

B. Hessmo and E. Sjöqvist, “Quantal phase for nonmaximally entangled photons,” Phys. Rev. A 62, 062301 (2000).
[CrossRef]

1999 (1)

1998 (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[CrossRef]

1993 (4)

S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[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]

1991 (1)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

1984 (1)

M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[CrossRef]

1959 (1)

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

1956 (1)

S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Abouraddy, A. F.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[CrossRef]

Abramochkin, E.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Aharonov, Y.

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Aiello, A.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Alfano, R. R.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[CrossRef]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Allen, L.

M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Andersen, U. L.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Aolita, L.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

L. Aolita and S. P. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
[CrossRef]

Banzer, P.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Berry, M. V.

M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[CrossRef]

Bohm, D.

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Borges, C. V. S.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

Boyd, R. W.

Cardano, F.

Castagnoli, G.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
[CrossRef]

Chen, L.

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

L. Chen and W. She, “Single-photon spin-orbit entanglement violating a Bell-like inequality,” J. Opt. Soc. Am. B 27, A7–A10 (2010).
[CrossRef]

Cirac, J. I.

L.-M. Duan, J. I. Cirac, and P. Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
[CrossRef]

Courtial, J.

Crawford, P. R.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

D’Ambrosio, V.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

Dandoloff, R.

R. Mosseri and R. Dandoloff, “Geometry of entangled states, Bloch spheres and Hopf fibrations,” J. Phys. A 34, 10243–10252 (2001).
[CrossRef]

de Lisio, C.

Di Giuseppe, G.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[CrossRef]

Du, J.

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

Duan, L.-M.

L.-M. Duan, J. I. Cirac, and P. Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
[CrossRef]

Ekert, A.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
[CrossRef]

Elser, D.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Ericsson, M.

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

Euser, T. G.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Evans, S.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[CrossRef]

Fortsch, M.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Franke-Arnold, S.

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

Gabriel, C.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Galvez, E. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Gori, F.

Haglin, P. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Hessmo, B.

B. Hessmo and E. Sjöqvist, “Quantal phase for nonmaximally entangled photons,” Phys. Rev. A 62, 062301 (2000).
[CrossRef]

Hor-Meyll, M.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[CrossRef]

Huguenin, J. A. O.

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[CrossRef]

Johansson, M.

M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
[CrossRef]

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

Joly, N. Y.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Jones, J. A.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
[CrossRef]

Kagalwala, K. H.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[CrossRef]

Karimi, E.

Khoury, A. Z.

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
[CrossRef]

L. E. Oxman and A. Z. Khoury, “Fractional topological phase for entangled qudits,” Phys. Rev. Lett. 106, 240503 (2011).
[CrossRef]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[CrossRef]

C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express 18, 9207–9212 (2010).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[CrossRef]

Kurzynowski, P.

Kwek, L. C.

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

Leach, J.

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

Leuchs, G.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Liao, C. J.

W. LiMing, Z. L. Tang, and C. J. Liao, “Representation of the SO(3) group by a maximally entangled state,” Phys. Rev. A 69, 064301 (2004).
[CrossRef]

LiMing, W.

W. LiMing, Z. L. Tang, and C. J. Liao, “Representation of the SO(3) group by a maximally entangled state,” Phys. Rev. A 69, 064301 (2004).
[CrossRef]

Machula, T.

Marquardt, C.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Marques, B.

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

Marrucci, L.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[CrossRef]

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

Matoso, A.

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

Milione, G.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[CrossRef]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Milman, P.

C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[CrossRef]

P. Milman, “Phase dynamics of entangled qubits,” Phys. Rev. A 73, 062118 (2006).
[CrossRef]

P. Milman and R. Mosseri, “Topological phase for entangled two-qubit states,” Phys. Rev. Lett. 90, 230403 (2003).
[CrossRef]

Mosseri, R.

P. Milman and R. Mosseri, “Topological phase for entangled two-qubit states,” Phys. Rev. Lett. 90, 230403 (2003).
[CrossRef]

R. Mosseri and R. Dandoloff, “Geometry of entangled states, Bloch spheres and Hopf fibrations,” J. Phys. A 34, 10243–10252 (2001).
[CrossRef]

Mukunda, N.

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[CrossRef]

Nagali, E.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

Nolan, D. A.

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[CrossRef]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

Oh, C. H.

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

Oxman, L. E.

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

L. E. Oxman and A. Z. Khoury, “Fractional topological phase for entangled qudits,” Phys. Rev. Lett. 106, 240503 (2011).
[CrossRef]

Padgett, M. J.

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

M. J. Padgett and J. Courtial, “Poincaré-sphere equivalent for light beams containing orbital angular momentum,” Opt. Lett. 24, 430–432 (1999).
[CrossRef]

Pádua, S.

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

Pancharatnam, S.

S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

S. Pancharatnam, Collected Works of S. Pancharatnam (Oxford University, 1975).

Peng, X.

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

Piccirillo, B.

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

Pysher, M. J.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Russell, P. S. J.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Saleh, B. E. A.

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[CrossRef]

Santamato, E.

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[CrossRef]

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

Schulz, S. A.

Sciarrino, F.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

She, W.

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

L. Chen and W. She, “Single-photon spin-orbit entanglement violating a Bell-like inequality,” J. Opt. Soc. Am. B 27, A7–A10 (2010).
[CrossRef]

Shi, M.

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Simon, R.

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

Singh, K.

M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
[CrossRef]

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

Sjöqvist, E.

M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
[CrossRef]

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

E. Sjöqvist, “Entanglement-induced geometric phase of quantum states,” Phys. Lett. A 374, 1431–1433 (2010).
[CrossRef]

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

B. Hessmo and E. Sjöqvist, “Quantal phase for nonmaximally entangled photons,” Phys. Rev. A 62, 062301 (2000).
[CrossRef]

E. Sjöqvist, “Geometric phase for entangled spin pairs,” Phys. Rev. A 62, 022109 (2000).
[CrossRef]

Slussarenko, S.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

F. Cardano, E. Karimi, S. Slussarenko, L. Marrucci, C. de Lisio, and E. Santamato, “Polarization pattern of vector vortex beams generated by q-plates with different topological charges,” Appl. Opt. 51, C1–C6 (2012).
[CrossRef]

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[CrossRef]

Souza, C. E. R.

C. E. R. Souza and A. Z. Khoury, “A Michelson controlled-not gate with a single-lens astigmatic mode converter,” Opt. Express 18, 9207–9212 (2010).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[CrossRef]

Spreeuw, R. J. C.

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[CrossRef]

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[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]

Suter, D.

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

Szarycz, M.

Sztul, H. I.

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Tang, Z. L.

W. LiMing, Z. L. Tang, and C. J. Liao, “Representation of the SO(3) group by a maximally entangled state,” Phys. Rev. A 69, 064301 (2004).
[CrossRef]

Tong, D. M.

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

van Enk, S. J.

S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

var der Veen, H. E. L. O.

M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vedral, V.

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
[CrossRef]

Volostnikov, V.

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

Walborn, S. P.

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

L. Aolita and S. P. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
[CrossRef]

Williams, R. E.

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Williamson, M. S.

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

Woerdman, J. P.

M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[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]

Wozniak, W. A.

Yariv, A.

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).

Yu, Y.-F.

Zhang, J.-Q.

Zhang, Z.-M.

Zhong, W.

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

Zhu, J.

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

Zoller, P.

L.-M. Duan, J. I. Cirac, and P. Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
[CrossRef]

Ann. Phys. (2)

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. I. General formalism,” Ann. Phys. 228, 205–268 (1993).
[CrossRef]

N. Mukunda and R. Simon, “Quantum kinematic approach to the geometric phase. II. The case of unitary group representations,” Ann. Phys. 228, 269–340 (1993).
[CrossRef]

Appl. Opt. (1)

Found. Phys. (1)

R. J. C. Spreeuw, “A classical analogy of entanglement,” Found. Phys. 28, 361–374 (1998).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (2)

J. Phys. A (1)

R. Mosseri and R. Dandoloff, “Geometry of entangled states, Bloch spheres and Hopf fibrations,” J. Phys. A 34, 10243–10252 (2001).
[CrossRef]

Nat. Commun. (1)

V. D’Ambrosio, E. Nagali, S. P. Walborn, L. Aolita, S. Slussarenko, L. Marrucci, and F. Sciarrino, “Complete experimental toolbox for alignment-free quantum communication,” Nat. Commun. 3, 961–968 (2012).
[CrossRef]

Nat. Photonics (1)

K. H. Kagalwala, G. Di Giuseppe, A. F. Abouraddy, and B. E. A. Saleh, “Bell’s measure in classical optical coherence,” Nat. Photonics 7, 72–78 (2013).
[CrossRef]

Nature (London) (1)

J. A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, “Geometric quantum computation using nuclear magnetic resonance,” Nature (London) 403, 869–871 (2000).
[CrossRef]

Opt. Commun. (3)

E. Abramochkin and V. Volostnikov, “Beam transformations and nontransformed beams,” Opt. Commun. 83, 123–135 (1991).
[CrossRef]

S. J. van Enk, “Geometric phase, transformations of Gaussian light beams and angular momentum transfer,” Opt. Commun. 102, 59–64 (1993).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. var der Veen, and J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Lett. A (1)

E. Sjöqvist, “Entanglement-induced geometric phase of quantum states,” Phys. Lett. A 374, 1431–1433 (2010).
[CrossRef]

Phys. Rev. (1)

Y. Aharonov and D. Bohm, “Significance of electromagnetic potentials in the quantum theory,” Phys. Rev. 115, 485–491 (1959).
[CrossRef]

Phys. Rev. A (14)

R. J. C. Spreeuw, “Classical wave-optics analogy of quantum-information processing,” Phys. Rev. A 63, 062302 (2001).
[CrossRef]

C. V. S. Borges, M. Hor-Meyll, J. A. O. Huguenin, and A. Z. Khoury, “Bell-like inequality for the spin-orbit separability of a laser beam,” Phys. Rev. A 82, 033833 (2010).
[CrossRef]

E. Karimi, J. Leach, S. Slussarenko, B. Piccirillo, L. Marrucci, L. Chen, W. She, S. Franke-Arnold, M. J. Padgett, and E. Santamato, “Spin-orbit hybrid entanglement of photons and quantum contextuality,” Phys. Rev. A 82, 022115 (2010).
[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]

E. Sjöqvist, “Geometric phase for entangled spin pairs,” Phys. Rev. A 62, 022109 (2000).
[CrossRef]

B. Hessmo and E. Sjöqvist, “Quantal phase for nonmaximally entangled photons,” Phys. Rev. A 62, 062301 (2000).
[CrossRef]

D. M. Tong, E. Sjöqvist, L. C. Kwek, C. H. Oh, and M. Ericsson, “Relation between geometric phases of entangled bipartite systems and their subsystems,” Phys. Rev. A 68, 022106 (2003).
[CrossRef]

C. E. R. Souza, C. V. S. Borges, A. Z. Khoury, J. A. O. Huguenin, L. Aolita, and S. P. Walborn, “Quantum key distribution without a shared reference frame,” Phys. Rev. A 77, 032345 (2008).
[CrossRef]

M. Johansson, M. Ericsson, K. Singh, E. Sjöqvist, and M. S. Williamson, “Topological phases and multiqubit entanglement,” Phys. Rev. A 85, 032112 (2012).
[CrossRef]

A. Z. Khoury, L. E. Oxman, B. Marques, A. Matoso, and S. Pádua, “Fractional topological phase on spatially encoded photonic qudits,” Phys. Rev. A 87, 042113 (2013).
[CrossRef]

M. Johansson, A. Z. Khoury, K. Singh, and E. Sjöqvist, “Three-qubit topological phase on entangled photon pairs,” Phys. Rev. A 87, 042112 (2013).
[CrossRef]

J. Du, J. Zhu, M. Shi, X. Peng, and D. Suter, “Experimental observation of a topological phase in the maximally entangled state of a pair of qubits,” Phys. Rev. A 76, 042121 (2007).
[CrossRef]

P. Milman, “Phase dynamics of entangled qubits,” Phys. Rev. A 73, 062118 (2006).
[CrossRef]

W. LiMing, Z. L. Tang, and C. J. Liao, “Representation of the SO(3) group by a maximally entangled state,” Phys. Rev. A 69, 064301 (2004).
[CrossRef]

Phys. Rev. Lett. (8)

C. Gabriel, A. Aiello, W. Zhong, T. G. Euser, N. Y. Joly, P. Banzer, M. Fortsch, D. Elser, U. L. Andersen, C. Marquardt, P. S. J. Russell, and G. Leuchs, “Entangling different degrees of freedom by quadrature squeezing cylindrically polarized modes,” Phys. Rev. Lett. 106, 060502 (2011).
[CrossRef]

L. E. Oxman and A. Z. Khoury, “Fractional topological phase for entangled qudits,” Phys. Rev. Lett. 106, 240503 (2011).
[CrossRef]

L. Aolita and S. P. Walborn, “Quantum communication without alignment using multiple-qubit single-photon states,” Phys. Rev. Lett. 98, 100501 (2007).
[CrossRef]

C. E. R. Souza, J. A. O. Huguenin, P. Milman, and A. Z. Khoury, “Topological phase for spin-orbit transformations on a laser beam,” Phys. Rev. Lett. 99, 160401 (2007).
[CrossRef]

G. Milione, H. I. Sztul, D. A. Nolan, and R. R. Alfano, “Higher-order Poincaré sphere, Stokes parameters, and the angular momentum of light,” Phys. Rev. Lett. 107, 053601 (2011).
[CrossRef]

G. Milione, S. Evans, D. A. Nolan, and R. R. Alfano, “Higher order Pancharatnam–Berry phase and the angular momentum of light,” Phys. Rev. Lett. 108, 190401 (2012).
[CrossRef]

P. Milman and R. Mosseri, “Topological phase for entangled two-qubit states,” Phys. Rev. Lett. 90, 230403 (2003).
[CrossRef]

E. J. Galvez, P. R. Crawford, H. I. Sztul, M. J. Pysher, P. J. Haglin, and R. E. Williams, “Geometric phase associated with mode transformations of optical beams bearing orbital angular momentum,” Phys. Rev. Lett. 90, 203901 (2003).
[CrossRef]

Proc. Indian Acad. Sci. A (1)

S. Pancharatnam, “Generalized theory of interference and its application,” Proc. Indian Acad. Sci. A 44, 247–262 (1956).

Proc. R. Soc. A (1)

M. V. Berry, “Quantal phase-factors accompanying adiabatic changes,” Proc. R. Soc. A 392, 45–57 (1984).
[CrossRef]

Science (1)

L.-M. Duan, J. I. Cirac, and P. Zoller, “Geometric manipulation of trapped ions for quantum computation,” Science 292, 1695–1697 (2001).
[CrossRef]

Other (3)

S. Pancharatnam, Collected Works of S. Pancharatnam (Oxford University, 1975).

A. E. Siegman, Lasers (University Science Books, 1986).

A. Yariv, Quantum Electronics, 3rd ed. (Wiley, 1988).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

Linear polarization patterns of cylindrically polarized beams.

Fig. 2.
Fig. 2.

Experimental setup. S-PLATE, radial polarization converter; QWP and HWP, quarter and half-wave plates, respectively; PBS and BS, polarizing and nonpolarizing beam splitters, respectively; AMC, astigmatic mode converter; GP, glass plate introduced to slightly modify the path of the beam. The orientation θ of HWP-2 is switched from 45° to +45° in order to perform 0- or π-class trajectories.

Fig. 3.
Fig. 3.

Diagram representing the transformations performed in the SO(3) sphere. The 0- class path is indicated in blue, and the π- class path between points 1 and 2 is indicated in white.

Fig. 4.
Fig. 4.

Images of the interference pattern observed at the output of the MZ interferometer. The corresponding values of ϵ and θ are indicated.

Fig. 5.
Fig. 5.

Density plot of Eq. (21), giving the theoretical images of the interference patterns at the output of the MZ interferometer. The images are in one-to-one correspondence with those in Fig. 4.

Equations (23)

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

ψ±(ρ,φ,z)=2πρw2(z)exp(ρ2w2(z))×exp{i[kρ22R(z)+2arctan(zzR)]}e±iφ,
ψh(x,y,z)=2π2xw2(z)exp(x2+y2w2(z))×exp{i[k(x2+y2)2R(z)+2arctan(zzR)]},
ψv(x,y,z)=2π2yw2(z)exp(x2+y2w2(z))×exp{i[k(x2+y2)2R(z)+2arctan(zzR)]}.
ψ±=ψh±iψv2.
ψ±45°=ψh±ψv2.
Ψ±=ψhe^H±ψve^V2,Φ±=ψhe^V±ψve^H2,
Ψ=λ1ψ+e^H+λ2ψe^H+λ3ψ+e^V+λ4ψe^V[λ1,λ2,λ3,λ4]T.
C=2|λ1λ4λ2λ3|.
W(θ,ϕ)=[cosϕ2+isinϕ2cos2θisinϕ2sin2θisinϕ2sin2θcosϕ2isinϕ2cos2θ].
C(θ,ϕ)=[cosϕ2isinϕ2e2iθisinϕ2e2iθcosϕ2].
Ψ0=ψ+e^H+ψe^V212[1,0,0,1]T,
Ψi=ϵψ+e^H+1ϵψe^V[ϵ,0,0,1ϵ]T,
Ψ1=[1C(22.5°,π/2)]Ψ0=12[1,e3iπ4,eiπ4,1]T.
Ψ2=[W(θ,π)1]Ψ1=12[icos2θsin2θeiπ4cos2θeiπ4+isin2θisin2θ+cos2θeiπ4sin2θeiπ4icos2θ].
Ψ3=[1C(22.5°,π/2)]Ψ2=12[sin2θeiπ4cos2θeiπ4cos2θeiπ4sin2θeiπ4],
Ψ4=[W(0°,π/2)1]Ψ3=12[sin2θicos2θicos2θsin2θ].
ΨNS=12(λψ+e^H+ηψe^Hη*ψ+e^V+λ*ψe^V)12[λ,η,η*,λ*]T.
λ=cosa2iuzsina2,η=(uy+iux)sina2.
Ψ0Ψ4(0,0,0,1),Ψ1(π2,12,12,0),Ψ2(2π3,23,0,13),Ψ3(π2,0,0,1).
ϕg=arg[Ψ0·Ψ]+i0TΨ·Ψ˙dt,
I(r)=|Ψ4(r)+eiq·rΨ0(r)|2=F(ρ,z)[12ϵ(1ϵ)sin2θcos(q·r)ϵcos2θsin(q·r+2φ)(1ϵ)cos2θsin(q·r2φ)],
F(ρ,z)=4ρ2πw4(z)exp(2ρ2w2(z))
V(θ=±45°)=|2ϵ(1ϵ)|.

Metrics