Abstract

We demonstrate that single photons from a rotating q-plate exhibit an entanglement in three degrees of freedom of spin, orbital angular momentum, and the rotational Doppler shift (RDS) due to the nonconservation of total spin and orbital angular momenta. We find that the rotational Doppler shift Δω=Ω(Δsl), where s, l and Ω are quantum numbers of spin, orbital angular momentum, and rotating velocity of the q-plate, respectively. Of interest is that the rotational Doppler shift directly reflects the rotational symmetry of q-plates and can be also expressed as Δωn, where n=2(q-1) denotes the fold number of rotational symmetry. Besides, based on this single-photon spin-orbit-RDS entanglement, we propose an experimental scheme to sort photons of different frequency shifts according to individual orbital angular momentum.

© 2008 Optical Society of America

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  1. L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
    [CrossRef]
  2. L. Allen, "Induction to the atoms and angular momentum of light special issue," J. Opt. B: Quantum Semiclass. Opt. 4, S1-S6 (2002).
    [CrossRef]
  3. M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004).
    [CrossRef]
  4. J. Courtial and K. O'Holleran, "Experiments with twisted light. Some of the mechanical and quantum-mechanical properties of optical vorteicest," Eur. Phys. J. Special Topics 145, 35-47 (2007).
    [CrossRef]
  5. G. Molina-Terriza, J. P. Torres, and L. Torner, "Twisted photons," Nature Phys. 3, 305-310 (2007).
    [CrossRef]
  6. M. J. Padgett, "Electromagnetism: Like a speeding watch," Nature 443, 924-925 (2006).
    [CrossRef] [PubMed]
  7. B. A. Garetz and S. Arnold, "Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate," Opt. Commun. 31, 1-3 (1979).
    [CrossRef]
  8. R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988).
    [CrossRef] [PubMed]
  9. F. Bretenaker and A. Le Floch, "Energy exchanges between a rotating retardation plate and a laser beam," Phys. Rev. Lett. 65, 2316 (1990).
    [CrossRef] [PubMed]
  10. L. Allen and M. Padgett, "Equivalent geometric transformations for spin and orbital angular momentum of light," J. Mod. Opt. 54, 487-491 (2007).
    [CrossRef]
  11. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 45, 8185-8189 (1992).
    [CrossRef] [PubMed]
  12. L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994).
    [CrossRef]
  13. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
    [CrossRef]
  14. J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
    [CrossRef]
  15. M. J. Padgett and J. Courtial, "Poincaré-sphere equivalent for light beams containing orbital angular momentum," Opt. Lett. 24, 430-432 (1999).
    [CrossRef]
  16. I. Bialynicki-Birula and Z. Bialynicka-Birula, "Rotational Frequency shift," Phys. Rev. Lett. 78, 2539-2542 (1997).
    [CrossRef]
  17. I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
    [CrossRef] [PubMed]
  18. S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
    [CrossRef] [PubMed]
  19. S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
    [CrossRef]
  20. N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998).
    [CrossRef]
  21. B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001).
    [CrossRef]
  22. Y.-H. Kim, "Single-photon two-qubit entangled states: Preparation and measurement," Phys. Rev. A 67, 040301 (2003).
    [CrossRef]
  23. A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
    [CrossRef]
  24. 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]
  25. L. Marrucci, C. Manzo, and D. Paparo, "Optical spin-to-orbital angular momentum conversion in inhomogeneous anisotropic media," Phys. Rev. Lett. 96, 163905 (2006).
    [CrossRef] [PubMed]
  26. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
    [CrossRef] [PubMed]
  27. E. Schmidt, "Zur Theorie der linearen und nicht linearen Integralgleichungen," Math. Ann. 63, 433-466 (1907).
    [CrossRef]
  28. Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
    [CrossRef] [PubMed]
  29. S. J. van Enk, "Single-particle entanglement," Phys. Rev. A 72, 064306 (2005).
    [CrossRef]
  30. S. J. van Enk, "Entanglement of electromagnetic fields," Phys. Rev. A 67, 022303 (2003).
    [CrossRef]
  31. L. Marrucci, "Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals," Proc. of SPIE 6587, 658708 (2007).
    [CrossRef]
  32. L. Deng, H. Wang, and K. Wang, "Quantum CNOT gates with orbital angular momentum and polarization of single-photon quantum logic," J. Opt. Soc. Am. B 24, 2517-2520 (2007).
    [CrossRef]
  33. L. Chen and W. She, "Electro-optically forbidden or enhanced spin-to-orbital angular momentum conversion in a focused light beam," Opt. Lett. 33, 696-698 (2008).
    [CrossRef] [PubMed]
  34. A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
    [CrossRef]
  35. R. J. C. Spreeuw, "A classical analogy of entanglement," Found. Phys. 28, 361-374 (1998).
    [CrossRef]
  36. M. J. Padgett, "The mechanism for energy transfer in the rotational frequency shift of a light beam," J. Opt. A: Pure Appl. Opt. 6, S263-S265 (2004).
    [CrossRef]
  37. I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
    [CrossRef]
  38. S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002).
    [CrossRef]
  39. S. C. McEldowney, D. M. Shemo, and R. A. Chipman, "Vortex retarders produced from photo-aligned liquid crystal polymers," Opt. Express 16, 7295-7308 (2008).
    [CrossRef] [PubMed]
  40. J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
    [CrossRef] [PubMed]

2008 (3)

2007 (6)

L. Deng, H. Wang, and K. Wang, "Quantum CNOT gates with orbital angular momentum and polarization of single-photon quantum logic," J. Opt. Soc. Am. B 24, 2517-2520 (2007).
[CrossRef]

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

L. Marrucci, "Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals," Proc. of SPIE 6587, 658708 (2007).
[CrossRef]

J. Courtial and K. O'Holleran, "Experiments with twisted light. Some of the mechanical and quantum-mechanical properties of optical vorteicest," Eur. Phys. J. Special Topics 145, 35-47 (2007).
[CrossRef]

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

L. Allen and M. Padgett, "Equivalent geometric transformations for spin and orbital angular momentum of light," J. Mod. Opt. 54, 487-491 (2007).
[CrossRef]

2006 (3)

M. J. Padgett, "Electromagnetism: Like a speeding watch," Nature 443, 924-925 (2006).
[CrossRef] [PubMed]

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
[CrossRef] [PubMed]

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

2005 (2)

S. J. van Enk, "Single-particle entanglement," Phys. Rev. A 72, 064306 (2005).
[CrossRef]

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

2004 (3)

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004).
[CrossRef]

M. J. Padgett, "The mechanism for energy transfer in the rotational frequency shift of a light beam," J. Opt. A: Pure Appl. Opt. 6, S263-S265 (2004).
[CrossRef]

2003 (4)

S. J. van Enk, "Entanglement of electromagnetic fields," Phys. Rev. A 67, 022303 (2003).
[CrossRef]

Y.-H. Kim, "Single-photon two-qubit entangled states: Preparation and measurement," Phys. Rev. A 67, 040301 (2003).
[CrossRef]

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
[CrossRef] [PubMed]

2002 (4)

L. Allen, "Induction to the atoms and angular momentum of light special issue," J. Opt. B: Quantum Semiclass. Opt. 4, S1-S6 (2002).
[CrossRef]

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
[CrossRef]

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002).
[CrossRef]

2001 (1)

B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001).
[CrossRef]

1999 (2)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

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

1998 (4)

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

N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

1997 (1)

I. Bialynicki-Birula and Z. Bialynicka-Birula, "Rotational Frequency shift," Phys. Rev. Lett. 78, 2539-2542 (1997).
[CrossRef]

1995 (1)

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

1994 (1)

L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994).
[CrossRef]

1992 (1)

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

1990 (1)

F. Bretenaker and A. Le Floch, "Energy exchanges between a rotating retardation plate and a laser beam," Phys. Rev. Lett. 65, 2316 (1990).
[CrossRef] [PubMed]

1988 (1)

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988).
[CrossRef] [PubMed]

1979 (1)

B. A. Garetz and S. Arnold, "Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate," Opt. Commun. 31, 1-3 (1979).
[CrossRef]

1907 (1)

E. Schmidt, "Zur Theorie der linearen und nicht linearen Integralgleichungen," Math. Ann. 63, 433-466 (1907).
[CrossRef]

Adami, C.

N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998).
[CrossRef]

Allen, L.

L. Allen and M. Padgett, "Equivalent geometric transformations for spin and orbital angular momentum of light," J. Mod. Opt. 54, 487-491 (2007).
[CrossRef]

M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004).
[CrossRef]

L. Allen, "Induction to the atoms and angular momentum of light special issue," J. Opt. B: Quantum Semiclass. Opt. 4, S1-S6 (2002).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994).
[CrossRef]

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

Arnold, S.

B. A. Garetz and S. Arnold, "Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate," Opt. Commun. 31, 1-3 (1979).
[CrossRef]

Babiker, M.

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994).
[CrossRef]

Badurek, G.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

Bakshaev, A. Ya.

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

Barnett, S. M.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002).
[CrossRef]

Baron, M.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

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]

Barreiro, S.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
[CrossRef] [PubMed]

Basistiy, I. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

Beige, A.

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
[CrossRef]

Beijersbergen, M. W.

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

Bekshaev, A. Ya.

Bialynicka-Birula, Z.

I. Bialynicki-Birula and Z. Bialynicka-Birula, "Rotational Frequency shift," Phys. Rev. Lett. 78, 2539-2542 (1997).
[CrossRef]

Bialynicki-Birula, I.

I. Bialynicki-Birula and Z. Bialynicka-Birula, "Rotational Frequency shift," Phys. Rev. Lett. 78, 2539-2542 (1997).
[CrossRef]

Bretenaker, F.

F. Bretenaker and A. Le Floch, "Energy exchanges between a rotating retardation plate and a laser beam," Phys. Rev. Lett. 65, 2316 (1990).
[CrossRef] [PubMed]

Cerf, N. J.

N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998).
[CrossRef]

Chen, L.

Chipman, R. A.

Courtial, J.

J. Courtial and K. O'Holleran, "Experiments with twisted light. Some of the mechanical and quantum-mechanical properties of optical vorteicest," Eur. Phys. J. Special Topics 145, 35-47 (2007).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004).
[CrossRef]

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

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

de Oliveira, A. N.

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Deng, L.

Dholakia, K.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

Englert, B. G.

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
[CrossRef]

B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001).
[CrossRef]

Failache, H.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
[CrossRef] [PubMed]

Franke-Arnold, S.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Friese, M. E. J.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Garetz, B. A.

B. A. Garetz and S. Arnold, "Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate," Opt. Commun. 31, 1-3 (1979).
[CrossRef]

Hasegawa, Y.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

He, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Heckenberg, N. R.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Kim, Y.-H.

Y.-H. Kim, "Single-photon two-qubit entangled states: Preparation and measurement," Phys. Rev. A 67, 040301 (2003).
[CrossRef]

Kimble, H. J.

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988).
[CrossRef] [PubMed]

Kurtsiefer, C.

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
[CrossRef]

B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001).
[CrossRef]

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]

N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998).
[CrossRef]

Le Floch, A.

F. Bretenaker and A. Le Floch, "Energy exchanges between a rotating retardation plate and a laser beam," Phys. Rev. Lett. 65, 2316 (1990).
[CrossRef] [PubMed]

Leach, J.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Lezama, A.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
[CrossRef] [PubMed]

Loidl, R.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

Manzo, C.

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

Marrucci, L.

L. Marrucci, "Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals," Proc. of SPIE 6587, 658708 (2007).
[CrossRef]

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

McEldowney, S. C.

Molina-Terriza, G.

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

Monken, C. H.

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Nienhuis, G.

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

O'Holleran, K.

J. Courtial and K. O'Holleran, "Experiments with twisted light. Some of the mechanical and quantum-mechanical properties of optical vorteicest," Eur. Phys. J. Special Topics 145, 35-47 (2007).
[CrossRef]

Padgett, M.

L. Allen and M. Padgett, "Equivalent geometric transformations for spin and orbital angular momentum of light," J. Mod. Opt. 54, 487-491 (2007).
[CrossRef]

Padgett, M. J.

M. J. Padgett, "Electromagnetism: Like a speeding watch," Nature 443, 924-925 (2006).
[CrossRef] [PubMed]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

M. J. Padgett, "The mechanism for energy transfer in the rotational frequency shift of a light beam," J. Opt. A: Pure Appl. Opt. 6, S263-S265 (2004).
[CrossRef]

M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004).
[CrossRef]

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

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

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

Paparo, D.

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

Power, W. L.

L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994).
[CrossRef]

Rauch, H.

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

Robertson, D. A.

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

Rubinsztein-Dunlop, H.

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

Schmidt, E.

E. Schmidt, "Zur Theorie der linearen und nicht linearen Integralgleichungen," Math. Ann. 63, 433-466 (1907).
[CrossRef]

She, W.

Shemo, D. M.

Simon, R.

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988).
[CrossRef] [PubMed]

Skeldon, K.

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

Slyusar, V. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

Soskin, M. S.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

Spreeuw, R. J. C.

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] [PubMed]

Sudarshan, E. C. G.

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988).
[CrossRef] [PubMed]

Tabosa, J. W. R.

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
[CrossRef] [PubMed]

Torner, L.

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

Torres, J. P.

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

van Enk, S. J.

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

S. J. van Enk, "Single-particle entanglement," Phys. Rev. A 72, 064306 (2005).
[CrossRef]

S. J. van Enk, "Entanglement of electromagnetic fields," Phys. Rev. A 67, 022303 (2003).
[CrossRef]

Vasnetsov, M. V.

I. V. Basistiy, V. V. Slyusar, M. S. Soskin, M. V. Vasnetsov, and A. Ya. Bekshaev, "Manifestation of the rotational Doppler effect by use of an off-axis optical vortex beam," Opt. Lett. 28, 1185-1187 (2003).
[CrossRef] [PubMed]

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

Walborn, S. P.

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

Wang, H.

Wang, K.

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]

Weinfurter, H.

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
[CrossRef]

B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001).
[CrossRef]

Woerdman, J. P.

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

Eur. Phys. J. Special Topics (1)

J. Courtial and K. O'Holleran, "Experiments with twisted light. Some of the mechanical and quantum-mechanical properties of optical vorteicest," Eur. Phys. J. Special Topics 145, 35-47 (2007).
[CrossRef]

Found. Phys. (1)

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

J. Mod. Opt. (1)

L. Allen and M. Padgett, "Equivalent geometric transformations for spin and orbital angular momentum of light," J. Mod. Opt. 54, 487-491 (2007).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

M. J. Padgett, "The mechanism for energy transfer in the rotational frequency shift of a light beam," J. Opt. A: Pure Appl. Opt. 6, S263-S265 (2004).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (3)

A. N. de Oliveira, S. P. Walborn, and C. H. Monken, "Implementing the Deutsch algorithm with polarization and transverse spatical modes," J. Opt. B: Quantum Semiclass. Opt. 7, 288-292 (2005).
[CrossRef]

S. M. Barnett, "Optical angular-momentum flux," J. Opt. B: Quantum Semiclass. Opt. 4, S7-S16 (2002).
[CrossRef]

L. Allen, "Induction to the atoms and angular momentum of light special issue," J. Opt. B: Quantum Semiclass. Opt. 4, S1-S6 (2002).
[CrossRef]

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

J. Phys. A: Math. Gen. (1)

A. Beige, B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Secure communication with single-photon two-qubit states," J. Phys. A: Math. Gen. 35, L407-L413 (2002).
[CrossRef]

JETP Lett. (1)

I. V. Basistiy, A. Ya. Bakshaev, M. V. Vasnetsov, V. V. Slyusar, and M. S. Soskin, "Observation of the rotational doppler effect for optical beams with helical wave front using spiral zone plate," JETP Lett. 76, 486-489 (2002).
[CrossRef]

Math. Ann. (1)

E. Schmidt, "Zur Theorie der linearen und nicht linearen Integralgleichungen," Math. Ann. 63, 433-466 (1907).
[CrossRef]

Nature (2)

Y. Hasegawa, R. Loidl, G. Badurek, M. Baron, and H. Rauch, "Violation of a Bell-like inequality in single-neutron interferometry," Nature 425, 45-48 (2003).
[CrossRef] [PubMed]

M. J. Padgett, "Electromagnetism: Like a speeding watch," Nature 443, 924-925 (2006).
[CrossRef] [PubMed]

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]

Opt. Commun. (2)

B. A. Garetz and S. Arnold, "Variable frequency shifting of circularly polarized laser radiation via a rotating half-wave retardation plate," Opt. Commun. 31, 1-3 (1979).
[CrossRef]

L. Allen, M. Babiker, and W. L. Power, "Azimuthal Doppler shift in light beams with orbital angular momentum," Opt. Commun. 112, 141-144 (1994).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. A (7)

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

S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007).
[CrossRef]

N. J. Cerf, C. Adami, and P. G. Kwiat, "Optical simulation of quantum logic," Phys. Rev. A 57, 1477-1480 (1998).
[CrossRef]

B. G. Englert, C. Kurtsiefer, and H. Weinfurter, "Universal unitary gate for single-photon two-qubit states," Phys. Rev. A 63, 032303 (2001).
[CrossRef]

Y.-H. Kim, "Single-photon two-qubit entangled states: Preparation and measurement," Phys. Rev. A 67, 040301 (2003).
[CrossRef]

S. J. van Enk, "Single-particle entanglement," Phys. Rev. A 72, 064306 (2005).
[CrossRef]

S. J. van Enk, "Entanglement of electromagnetic fields," Phys. Rev. A 67, 022303 (2003).
[CrossRef]

Phys. Rev. Lett. (9)

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

H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995).
[CrossRef] [PubMed]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, and M. J. Padgett, "Measurement of the rotational frequency shift imparted to a rotating light beams possessing orbital angular momentum," Phys. Rev. Lett. 80, 3217-3219 (1998).
[CrossRef]

J. Courtial, D. A. Robertson, K. Dholakia, L. Allen, and M. J. Padgett, "Rotational frequency shift of a light beam," Phys. Rev. Lett. 81, 4828-4830 (1998).
[CrossRef]

R. Simon, H. J. Kimble, and E. C. G. Sudarshan, "Evolving geometric phase and its dynamical manifestation as a frequency shift: an optical experiment," Phys. Rev. Lett. 61, 19-22 (1988).
[CrossRef] [PubMed]

F. Bretenaker and A. Le Floch, "Energy exchanges between a rotating retardation plate and a laser beam," Phys. Rev. Lett. 65, 2316 (1990).
[CrossRef] [PubMed]

I. Bialynicki-Birula and Z. Bialynicka-Birula, "Rotational Frequency shift," Phys. Rev. Lett. 78, 2539-2542 (1997).
[CrossRef]

J. Leach, J. Courtial, K. Skeldon, S. M. Barnett, S. Franke-Arnold, and M. J. Padgett, "Interferometric methods to measure orbital and spin, or the total angular momentum of a single photon," Phys. Rev. Lett. 92, 013601 (2004).
[CrossRef] [PubMed]

S. Barreiro, J. W. R. Tabosa, H. Failache, and A. Lezama, "Spectroscopic observation of the rotational Doppler effect," Phys. Rev. Lett. 97, 113601 (2006).
[CrossRef] [PubMed]

Phys. Today (1)

M. J. Padgett, J. Courtial, and L. Allen, "Light??s orbital angular momentum," Phys. Today 57, 35-40 (2004).
[CrossRef]

Proc. of SPIE (1)

L. Marrucci, "Rotating light with light: Generation of helical modes of light by spin-to-orbital angular momentum conversion in inhomogeneous liquid crystals," Proc. of SPIE 6587, 658708 (2007).
[CrossRef]

Prog. Opt. (1)

L. Allen, M. J. Padgett, and M. Babiker, "The orbital angular momentum of light," Prog. Opt. 39, 291-372 (1999).
[CrossRef]

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Figures (3)

Fig. 1.
Fig. 1.

Illustration of some types of q-plates, in which the orientation of optical axis is the locally tangent to the line: (a) is rotational invariance; while (b), (c), (d), and (e) are of one-, two-, three-, and four-fold rotational symmetry, respectively.

Fig. 2.
Fig. 2.

The experimental scheme using q-plate to generate single-photon spin-orbit-RDS entanglement; and sort photons of different RDS according to individual OAM: BS, 50:50 non-polarizing beam splitter; DP, Dove prism; CBS, circularly polarizing beam splitter; M, mirror.

Fig. 3.
Fig. 3.

Phase diagrams of photons with OAM (a) l=-1, (b) l=0, and (c) l=+1, respectively.

Tables (1)

Tables Icon

Table 1. The example input states and q against resulting frequency shifts.

Equations (3)

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

T q = 1 + exp ( i χ ) 2 [ 1 0 0 1 ] + 1 exp ( i χ ) ] 2 [ 0 exp [ i 2 α ( r , φ ) ] exp [ i 2 α ( r , φ ) ] 0 ] .
| out = α 1 | in , m + α 2 exp [ i 2 ( q 1 ) Ω t ] | L , m + 2 q + α 3 exp [ i 2 ] ( q 1 ) Ω t | R , m 2 q .
out = α 1 in , m Δ ω = 0 + α 2 L , m + 2 q Δ ω = n Ω + α 3 R , m 2 q Δ ω = n Ω .

Metrics