Abstract

We report on the amplification of an optical vortex beam carrying orbital angular momentum via induced narrow Raman gain in an ensemble of cold cesium atoms. A 20% single-pass Raman gain of a weak vortex signal field is observed with a spectral width of order of 1 MHz, much smaller than the natural width, demonstrating that the amplification process preserves the phase structure of the vortex beam. The gain is observed in the degenerated two-level system associated with the hyperfine transition 6S1/2(F = 3) ↔ 6P3/2(F′ = 2) of cesium. Our experimental observations are explained with a simple theoretical model based on a three-level Λ system interacting coherently with the weak Laguerre-Gauss field and a strong coupling field, including an incoherent pumping rate between the two degenerate ground-states.

© 2016 Optical Society of America

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References

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  1. M. Padgett, J. Courtial, and L. Allen, ”Light’s orbital angular momentum,” Phys. Today 57, 35 (2004).
    [Crossref]
  2. Juan P. Torres and Lluis Torner, Twisted Photons: Application of Light with Orbital Angular Momentum (Wiley-VCH, 2011).
    [Crossref]
  3. H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
    [Crossref]
  4. Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
    [Crossref] [PubMed]
  5. A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
    [Crossref]
  6. 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 (1992).
    [Crossref] [PubMed]
  7. R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
    [Crossref] [PubMed]
  8. D. Moretti, D. Felinto, and J. W. R. Tabosa, ”Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
    [Crossref]
  9. L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, ”Reversible optical memory for twisted photons,” Opt. Lett. 38, 712 (2013).
    [Crossref] [PubMed]
  10. R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
    [Crossref]
  11. R. A. de Oliveira, G. C. Borba, W. S. Martins, S. Barreiro, D. Felinto, and J. W. R. Tabosa, ”Nonlinear optical memory for manipulation of orbital angular momentum of light,” Opt. Lett. 40, 4939 (2015).
    [Crossref] [PubMed]
  12. S. Barreiro and J. W. R. Tabosa, ”Generation of light carrying orbital angular momentum via induced coherent grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
    [Crossref]
  13. W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
    [Crossref]
  14. A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
    [Crossref] [PubMed]
  15. P. Kumar and J. H. Shapiro, ”Observation of Raman-shifted oscillation near the sodium D-lines,” Opt. Lett. 10, 226 (1985).
    [Crossref] [PubMed]
  16. J. L. Bowie, J. C. Garrison, and R. Y. Chiao, ”Stimulated Raman gain in a Λ-type atomic system with doubly excited transitions,” Phys. Rev. A 61, 053811 (2000).
    [Crossref]
  17. G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
    [Crossref] [PubMed]
  18. Y. Zhu and J. Lin, ”Sub-Doppler light amplification in a coherently pumped atomic system,” Phys. Rev. A 53, 1767 (1996).
    [Crossref] [PubMed]
  19. J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
    [Crossref] [PubMed]
  20. D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
    [Crossref]
  21. A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
    [Crossref]
  22. A. Y. Bekshaev, M. Soskin, and M. Vasnetsov, ”Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237 (2004).
    [Crossref]
  23. P. Vaity, J. Banerji, and R. P. Singh, ”Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154 (2013).
    [Crossref]

2015 (2)

2014 (2)

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

2013 (2)

P. Vaity, J. Banerji, and R. P. Singh, ”Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154 (2013).
[Crossref]

L. Veissier, A. Nicolas, L. Giner, D. Maxein, A. S. Sheremet, E. Giacobino, and J. Laurat, ”Reversible optical memory for twisted photons,” Opt. Lett. 38, 712 (2013).
[Crossref] [PubMed]

2009 (1)

D. Moretti, D. Felinto, and J. W. R. Tabosa, ”Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

2008 (1)

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

2007 (1)

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

2006 (1)

W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
[Crossref]

2004 (2)

A. Y. Bekshaev, M. Soskin, and M. Vasnetsov, ”Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237 (2004).
[Crossref]

M. Padgett, J. Courtial, and L. Allen, ”Light’s orbital angular momentum,” Phys. Today 57, 35 (2004).
[Crossref]

2003 (1)

S. Barreiro and J. W. R. Tabosa, ”Generation of light carrying orbital angular momentum via induced coherent grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

2000 (2)

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

J. L. Bowie, J. C. Garrison, and R. Y. Chiao, ”Stimulated Raman gain in a Λ-type atomic system with doubly excited transitions,” Phys. Rev. A 61, 053811 (2000).
[Crossref]

1999 (1)

A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
[Crossref]

1996 (2)

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Y. Zhu and J. Lin, ”Sub-Doppler light amplification in a coherently pumped atomic system,” Phys. Rev. A 53, 1767 (1996).
[Crossref] [PubMed]

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 (1992).
[Crossref] [PubMed]

1991 (2)

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

1985 (1)

Ahmed, N.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Akulshin, A. M.

A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
[Crossref]

Allen, L.

M. Padgett, J. Courtial, and L. Allen, ”Light’s orbital angular momentum,” Phys. Today 57, 35 (2004).
[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 (1992).
[Crossref] [PubMed]

Banerji, J.

P. Vaity, J. Banerji, and R. P. Singh, ”Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154 (2013).
[Crossref]

Bao, C.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Barbosa, P. S.

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

Barreiro, S.

R. A. de Oliveira, G. C. Borba, W. S. Martins, S. Barreiro, D. Felinto, and J. W. R. Tabosa, ”Nonlinear optical memory for manipulation of orbital angular momentum of light,” Opt. Lett. 40, 4939 (2015).
[Crossref] [PubMed]

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

S. Barreiro and J. W. R. Tabosa, ”Generation of light carrying orbital angular momentum via induced coherent grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
[Crossref]

Bechmann-Pasquinucci, H.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[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 (1992).
[Crossref] [PubMed]

Bekshaev, A. Y.

A. Y. Bekshaev, M. Soskin, and M. Vasnetsov, ”Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237 (2004).
[Crossref]

Bloch, D.

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

Borba, G. C.

Bowie, J. L.

J. L. Bowie, J. C. Garrison, and R. Y. Chiao, ”Stimulated Raman gain in a Λ-type atomic system with doubly excited transitions,” Phys. Rev. A 61, 053811 (2000).
[Crossref]

Boyd, R. W.

Boyer, V.

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

Cao, Y.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Chen, G.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

Chen, Q.-F.

W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
[Crossref]

Chiao, R. Y.

J. L. Bowie, J. C. Garrison, and R. Y. Chiao, ”Stimulated Raman gain in a Λ-type atomic system with doubly excited transitions,” Phys. Rev. A 61, 053811 (2000).
[Crossref]

Courtial, J.

M. Padgett, J. Courtial, and L. Allen, ”Light’s orbital angular momentum,” Phys. Today 57, 35 (2004).
[Crossref]

Courtois, J.-Y.

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

Davidson, N.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

de Oliveira, R. A.

R. A. de Oliveira, G. C. Borba, W. S. Martins, S. Barreiro, D. Felinto, and J. W. R. Tabosa, ”Nonlinear optical memory for manipulation of orbital angular momentum of light,” Opt. Lett. 40, 4939 (2015).
[Crossref] [PubMed]

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

Felinto, D.

R. A. de Oliveira, G. C. Borba, W. S. Martins, S. Barreiro, D. Felinto, and J. W. R. Tabosa, ”Nonlinear optical memory for manipulation of orbital angular momentum of light,” Opt. Lett. 40, 4939 (2015).
[Crossref] [PubMed]

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

D. Moretti, D. Felinto, and J. W. R. Tabosa, ”Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

Firstenberg, O.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

Fry, E. S.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Garrison, J. C.

J. L. Bowie, J. C. Garrison, and R. Y. Chiao, ”Stimulated Raman gain in a Λ-type atomic system with doubly excited transitions,” Phys. Rev. A 61, 053811 (2000).
[Crossref]

Giacobino, E.

Giner, L.

Grison, D.

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

Grynberg, G.

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

Guo, G.-C.

W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
[Crossref]

Hu, Z.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

Huang, H.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Jiang, W.

W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
[Crossref]

Jones, K. M.

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

Kimble, H. J.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

Kumar, P.

Laurat, J.

Lavery, M. P. J.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Lee, R. B.

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

Lemons, K.

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

Lett, P. D.

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

Lezama, A.

A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
[Crossref]

Li, L.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Liao, P.

Lin, J.

Y. Zhu and J. Lin, ”Sub-Doppler light amplification in a coherently pumped atomic system,” Phys. Rev. A 53, 1767 (1996).
[Crossref] [PubMed]

Lipsich, A.

A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
[Crossref]

Liu, C.

Lounis, B.

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

Lukin, M. D.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Marino, A. M.

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

Martins, W. S.

R. A. de Oliveira, G. C. Borba, W. S. Martins, S. Barreiro, D. Felinto, and J. W. R. Tabosa, ”Nonlinear optical memory for manipulation of orbital angular momentum of light,” Opt. Lett. 40, 4939 (2015).
[Crossref] [PubMed]

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

Maxein, D.

Mirhosseini, M.

Molisch, A. F.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Moretti, D.

D. Moretti, D. Felinto, and J. W. R. Tabosa, ”Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

Nicolas, A.

Nikonov, D. E.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Padgett, M.

M. Padgett, J. Courtial, and L. Allen, ”Light’s orbital angular momentum,” Phys. Today 57, 35 (2004).
[Crossref]

Padgett, M. J.

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Padmabandu, G. G.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Pooser, R. C.

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

Pruvost, L.

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

Pugatch, R.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

Ren, Y.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Ron, A.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

Salomon, C.

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

Scully, M. O.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Shapiro, J. H.

Sheremet, A. S.

Shubin, I. N.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Shuker, M.

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

Singh, R. P.

P. Vaity, J. Banerji, and R. P. Singh, ”Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154 (2013).
[Crossref]

Soskin, M.

A. Y. Bekshaev, M. Soskin, and M. Vasnetsov, ”Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237 (2004).
[Crossref]

Spreeuw, R. J. C.

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

Tabosa, J. W. R.

R. A. de Oliveira, G. C. Borba, W. S. Martins, S. Barreiro, D. Felinto, and J. W. R. Tabosa, ”Nonlinear optical memory for manipulation of orbital angular momentum of light,” Opt. Lett. 40, 4939 (2015).
[Crossref] [PubMed]

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

D. Moretti, D. Felinto, and J. W. R. Tabosa, ”Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

S. Barreiro and J. W. R. Tabosa, ”Generation of light carrying orbital angular momentum via induced coherent grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

Tittel, W.

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Torner, Lluis

Juan P. Torres and Lluis Torner, Twisted Photons: Application of Light with Orbital Angular Momentum (Wiley-VCH, 2011).
[Crossref]

Torres, Juan P.

Juan P. Torres and Lluis Torner, Twisted Photons: Application of Light with Orbital Angular Momentum (Wiley-VCH, 2011).
[Crossref]

Tur, M.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Vaity, P.

P. Vaity, J. Banerji, and R. P. Singh, ”Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154 (2013).
[Crossref]

Vasnetsov, M.

A. Y. Bekshaev, M. Soskin, and M. Vasnetsov, ”Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237 (2004).
[Crossref]

Veissier, L.

Wang, Z.

Welch, G. R.

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

Willner, A. E.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Wilnner, A. J.

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 (1992).
[Crossref] [PubMed]

Xie, G.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Yan, Y.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Zhang, Y.-S.

W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
[Crossref]

Zhao, Z.

A. J. Wilnner, Y. Ren, G. Xie, Z. Zhao, Y. Cao, L. Li, N. Ahmed, Z. Wang, Y. Yan, P. Liao, C. Liu, M. Mirhosseini, R. W. Boyd, M. Tur, and A. E. Willner, ”Experimental demonstration of 20 Gbit/s data encoding and 2 ns channel hopping using orbital angular momentum modes,” Opt. Lett. 40, 5810 (2015).
[Crossref]

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Zhu, Y.

Y. Zhu and J. Lin, ”Sub-Doppler light amplification in a coherently pumped atomic system,” Phys. Rev. A 53, 1767 (1996).
[Crossref] [PubMed]

Appl. Phys. B (1)

R. A. de Oliveira, L. Pruvost, P. S. Barbosa, W. S. Martins, S. Barreiro, D. Felinto, D. Bloch, and J. W. R. Tabosa, ”Off-axis retrieval of orbital angular momentum of light stored in cold atoms,” Appl. Phys. B 117, 1121 (2014).
[Crossref]

Europhys. Lett. (1)

D. Grison, B. Lounis, C. Salomon, J.-Y. Courtois, and G. Grynberg, ”Raman-spectroscopy of cesium atoms in a laser trap,” Europhys. Lett. 15, 149 (1991).
[Crossref]

Nat. Commun. (1)

Y. Yan, G. Xie, M. P. J. Lavery, H. Huang, N. Ahmed, C. Bao, Y. Ren, Y. Cao, L. Li, Z. Zhao, A. F. Molisch, M. Tur, M. J. Padgett, and A. E. Willner, ”High-capacity millimetre-wave communications with orbital angular momentum multiplexing,” Nat. Commun. 5, 4876 (2014).
[Crossref] [PubMed]

Opt. Commun. (1)

A. Y. Bekshaev, M. Soskin, and M. Vasnetsov, ”Transformation of higher-order optical vortices upon focusing by an astigmatic lens,” Opt. Commun. 241, 237 (2004).
[Crossref]

Opt. Lett. (4)

Phys. Lett. A (1)

P. Vaity, J. Banerji, and R. P. Singh, ”Measuring the topological charge of an optical vortex by using a tilted convex lens,” Phys. Lett. A 377, 1154 (2013).
[Crossref]

Phys. Rev. A (7)

A. Lezama, S. Barreiro, A. Lipsich, and A. M. Akulshin, ”Coherent two-field spectroscopy of degenerate two-level systems,” Phys. Rev. A 61, 013801 (1999).
[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 (1992).
[Crossref] [PubMed]

D. Moretti, D. Felinto, and J. W. R. Tabosa, ”Collapses and revivals of stored orbital angular momentum of light in a cold-atom ensemble,” Phys. Rev. A 79, 023825 (2009).
[Crossref]

J. L. Bowie, J. C. Garrison, and R. Y. Chiao, ”Stimulated Raman gain in a Λ-type atomic system with doubly excited transitions,” Phys. Rev. A 61, 053811 (2000).
[Crossref]

Y. Zhu and J. Lin, ”Sub-Doppler light amplification in a coherently pumped atomic system,” Phys. Rev. A 53, 1767 (1996).
[Crossref] [PubMed]

W. Jiang, Q.-F. Chen, Y.-S. Zhang, and G.-C. Guo, ”Computation of topological charges of optical vortices via nondegenerate four-wave mixing,” Phys. Rev. A 74, 043811 (2006).
[Crossref]

H. Bechmann-Pasquinucci and W. Tittel, “Quantum cryptography using larger alphabets,” Phys. Rev. A 61, 062308 (2000).
[Crossref]

Phys. Rev. Lett. (5)

A. M. Marino, V. Boyer, R. C. Pooser, P. D. Lett, K. Lemons, and K. M. Jones, ”Delocalized correlations in twin light beams with orbital angular momentum,” Phys. Rev. Lett. 101, 093602 (2008).
[Crossref] [PubMed]

J. W. R. Tabosa, G. Chen, Z. Hu, R. B. Lee, and H. J. Kimble, ”Nonlinear spectroscopy of cold atoms in a spontaneous-force optical trap,” Phys. Rev. Lett. 66, 3245 (1991).
[Crossref] [PubMed]

G. G. Padmabandu, G. R. Welch, I. N. Shubin, E. S. Fry, D. E. Nikonov, M. D. Lukin, and M. O. Scully, ”Laser oscillation without population inversion in a sodium atomic beam,” Phys. Rev. Lett. 76, 2053 (1996).
[Crossref] [PubMed]

R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, ”Topological stability of stored optical vortices,” Phys. Rev. Lett. 98, 203601 (2007).
[Crossref] [PubMed]

S. Barreiro and J. W. R. Tabosa, ”Generation of light carrying orbital angular momentum via induced coherent grating in cold atoms,” Phys. Rev. Lett. 90, 133001 (2003).
[Crossref]

Phys. Today (1)

M. Padgett, J. Courtial, and L. Allen, ”Light’s orbital angular momentum,” Phys. Today 57, 35 (2004).
[Crossref]

Other (1)

Juan P. Torres and Lluis Torner, Twisted Photons: Application of Light with Orbital Angular Momentum (Wiley-VCH, 2011).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1

(a) Simplified experimental scheme showing the propagation directions of the incident beams (C, S and P). The direction of beam P is chosen to eliminate the FWM parametric gain. (b) The degenerate two-level system of cesium, corresponding to the hyperfine transition (F = 3) ↔ (F′ = 2), showing a generic set of Zeeman sublevels and indicating the coupling of the sublevels with the respective optical fields.

Fig. 2
Fig. 2

(a) Signal beam transmission as a function of the frequency detuning ΔS for different values of the detuning ΔC. The beam P is blue-detuned by ΔP = 1MHz and the beams intensities are indicated in the text. (b) Normalized signal beam transmission spectrum calculated for three different detunings of the coupling beam as indicated. The transmitted signal is normalized by the incident signal beam intensity. The used parameters, in units of Γ, are: ΩC = 0.1, ΓP = 0.05

Fig. 3
Fig. 3

Simplified Λ three-level scheme interacting coherently with the beams C and S. In the theoretical model we account for the beam P through a directional incoherent pumping rate ΓP from state |c〉 to state |a〉.

Fig. 4
Fig. 4

Measurement of the topological charge of the signal beam S. First column: Topological charge of the incident beam, measured without the atoms. Second column: Topological charge of the amplified signal beam via the Raman gain mechanism. Third column: Transversal profile of the images shown in the first column (red) and in the second column (black). For these data, the beams C and S are on Raman resonance and red-detuned by 6.8 MHz, while the P beam is on resonance.

Equations (12)

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

σ ˙ a a = i [ Ω C σ b a Ω C * σ a b ] + Γ b a σ b b + Γ P σ c c ,
σ ˙ c c = i [ Ω S σ b c Ω S * σ c b ] + Γ b c σ b b Γ P σ c c ,
σ ˙ b b = i [ Ω C σ b a Ω C * σ a b + Ω S σ b c Ω S * σ c b ] Γ σ b b ,
σ ˙ a b = i Δ C σ a b + i Ω C ( σ b b σ a a ) i Ω S σ a c Γ 2 σ a b ,
σ ˙ c b = i Δ C σ c b + i Ω S ( σ b b σ c c ) i Ω C σ c a Γ 2 σ c b ,
σ ˙ a c = i δ σ a c + i [ Ω C σ b c Ω S * σ a b ] Γ P 2 σ a c .
σ c b ( 1 ) = Ω S [ δ + i Γ P 2 ] ( σ c c ( 0 ) σ b b ( 0 ) ) Ω C Ω S σ b a ( 0 ) [ i δ + Γ P 2 ] ( i Δ S Γ 2 ) Ω C 2
σ a a ( 0 ) = Ω C 2 + Δ C 2 + Γ 2 / 4 Ω C 2 ( 2 + Γ / 2 Γ P ) + Δ C 2 + Γ 2 / 4 ,
σ b b ( 0 ) = 1 σ a a ( 0 ) 1 + Γ / 2 Γ P ,
σ c c ( 0 ) = Γ 2 Γ P 1 σ a a ( 0 ) 1 + Γ / 2 Γ P ,
σ a b ( 0 ) = i Ω C 1 ( 2 + Γ / 2 Γ P ) σ a a ( 0 ) ( i Δ C Γ / 2 ) ( 1 + Γ / 2 Γ P ) .
S ( r ) = L G p ( ρ , ϕ ) = 0 ( ρ 2 w 0 ) | | e ρ 2 w 0 e i ϕ L p | | ( 2 ρ 2 w 0 2 ) ,

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