Abstract

Recently a scheme has been proposed for detection of the structured light by measuring the transmission of a vortex beam through a cloud of cold rubidium atoms with energy levels of the Λ-type configuration [N. Radwell et al., Phys. Rev. Lett. 114, 123603 (2015) ]. This enables observation of regions of spatially dependent electromagnetically induced transparency (EIT). Here we suggest another scenario for detection of the structured light by measuring the absorption profile of a weak nonvortex probe beam in a highly resonant five-level combined tripod and Λ (CTL) atom-light coupling setup. We demonstrate that due to the closed-loop structure of CTL scheme, the absorption of the probe beam depends on the azimuthal angle and orbital angular momentum (OAM) of the control vortex beams. This feature is missing in simple Λ or tripod schemes, as there is no loop in such atom-light couplings. One can identify different regions of spatially structured transparency through measuring the absorption of probe field under different configurations of structured control light.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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  38. D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Linear up-conversion of orbital angular momentum,” Opt. Lett. 37, 3270–3272 (2012).
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  46. N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
    [Crossref] [PubMed]
  47. F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the hanle effect of the d1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997).
    [Crossref]
  48. L. Li, H. Guo, F. Xiao, X. Peng, and X. Chen, “Control of light in an m-type five-level atomic system,” J. Opt. Soc. Am. B 22, 1309–1313 (2005).
    [Crossref]
  49. Y. Hong, Y. Dong, Z. Mei, F. Bo, Z. Yan, and W. Jin-Hui, “Absorption and dispersion control in a five-level m-type atomic system,” Chin. Phys. B 21, 114207 (2012).
    [Crossref]
  50. J. Sheng, X. Yang, U. Khadka, and M. Xiao, “All-optical switching in an n-type four-level atom-cavity system,” Opt. Express 19, 17059–17064 (2011).
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  51. X. Yang, K. Ying, Y. Niu, and S. Gong, “Reversible self-kerr nonlinearity in an n-type atomic system through a switching field,” J. Opt. 17, 045505 (2015).
    [Crossref]

2017 (3)

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19, 043001 (2017).
[Crossref] [PubMed]

H. R. Hamedi, J. Ruseckas, and G. Juzeliūnas, “Electromagnetically induced transparency and nonlinear pulse propagation in a combined tripod and lambda atom-light coupling scheme,” J. Phys. B: At. Mol. Opt. Phys. 50, 185401 (2017).
[Crossref]

S. Shi, D.-S. Ding, W. Zhang, Z.-Y. Zhou, M.-X. Dong, S.-L. Liu, K. Wang, B.-S. Shi, and G.-C. Guo, “Transverse azimuthal dephasing of a vortex spin wave in a hot atomic gas,” Phys. Rev. A 95, 033823 (2017).
[Crossref]

2016 (2)

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 kilometers,” Proc. Natl Acad. Sci. 113, 13648 (2016).
[Crossref]

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl Acad. Sci. 113, 13642 (2016).
[Crossref]

2015 (5)

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
[Crossref] [PubMed]

A. M. Akulshin, R. J. McLean, E. E. Mikhailov, and I. Novikova, “Distinguishing nonlinear processes in atomic media via orbital angular momentum transfer,” Opt. Lett. 40, 1109–1112 (2015).
[Crossref] [PubMed]

X. Yang, K. Ying, Y. Niu, and S. Gong, “Reversible self-kerr nonlinearity in an n-type atomic system through a switching field,” J. Opt. 17, 045505 (2015).
[Crossref]

T. G. Akin, S. P. Krzyzewski, A. M. Marino, and E. R. I. Abraham, “Electromagnetically induced transparency with laguerre-gaussian modes in ultracold rubidium,” Opt. Commun. 339, 209 (2015).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

2014 (1)

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriasov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

2013 (3)

S. R. Chanu and V. Natarajan, “Narrowing of resonances in electromagnetically induced transparency and absorption using a laguerre-gaussian control beam,” Opt. Commun. 295, 150 (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]

J. Ruseckas, V. Kudriašov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
[Crossref]

2012 (3)

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, “Identifying the orbital angular momentum of light based on atomic ensembles,” Eur. Lett. 99, 34003 (2012).
[Crossref]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Linear up-conversion of orbital angular momentum,” Opt. Lett. 37, 3270–3272 (2012).
[Crossref] [PubMed]

Y. Hong, Y. Dong, Z. Mei, F. Bo, Z. Yan, and W. Jin-Hui, “Absorption and dispersion control in a five-level m-type atomic system,” Chin. Phys. B 21, 114207 (2012).
[Crossref]

2011 (3)

J. Sheng, X. Yang, U. Khadka, and M. Xiao, “All-optical switching in an n-type four-level atom-cavity system,” Opt. Express 19, 17059–17064 (2011).
[Crossref] [PubMed]

J. Ruseckas, A. Mekys, and G. Juzeliūnas, “Slow polaritons with orbital angular momentum in atomic gases,” Phys. Rev. A 83, 023812 (2011).
[Crossref]

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161 (2011).
[Crossref]

2010 (1)

R. G. Unanyan, J. Otterbach, M. Fleischhauer, J. Ruseckas, V. Kudriašov, and G. Juzeliūnas, “Spinor slow-light and dirac particles with variable mass,” Phys. Rev. Lett. 105, 173603 (2010).
[Crossref]

2009 (1)

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

2007 (2)

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]

D. Bhattacharyya, B. Ray, and P. N. Ghosh, “Theoretical study of electromagnetically induced transparency in a five-level atom and application to doppler-broadened and doppler-free rb atoms,” J. Phys. B: At. Mol. Opt. Phys. 40, 4061 (2007).
[Crossref]

2006 (1)

M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive kerr nonlinearity: Theory,” Phys. Rev. A 74, 013812 (2006).
[Crossref]

2005 (6)

M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837 (2005).
[Crossref] [PubMed]

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature 438, 833 (2005).
[Crossref] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

A. Andre, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin, “Quantum control of light using electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, S589–S604 (2005).
[Crossref]

L. Li, H. Guo, F. Xiao, X. Peng, and X. Chen, “Control of light in an m-type five-level atomic system,” J. Opt. Soc. Am. B 22, 1309–1313 (2005).
[Crossref]

2004 (1)

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

2003 (3)

D. McGloin, “Coherent effects in a driven vee scheme,” J. Phys. B: At. Mol. Opt. Phys. 36, 2861 (2003).
[Crossref]

M. D. Lukin, “Colloquium: Trapping and manipulating photon states in atomic ensembles,” Rev. Mod. Phys. 75, 457 (2003).
[Crossref]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638 (2003).
[Crossref] [PubMed]

2002 (2)

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[Crossref]

E. Paspalakis and P. L. Knight, “Transparency, slow light and enhanced nonlinear optics in a four-level scheme,” J. Opt. B: Quantum Semiclass. Opt. 4, S372 (2002).
[Crossref]

2001 (4)

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

D. McGloin, D. J. Fulton, and M. H. Dunn, “Electromagnetically induced transparency in n-level cascade schemes,” Opt. Commun. 190, 221 (2001).
[Crossref]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783 (2001).
[Crossref] [PubMed]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490 (2001).
[Crossref] [PubMed]

2000 (1)

M. Fleischhauer and M. D. Lukin, “Dark-state polaritons in electromagnetically induced transparency,” Phys. Rev. Lett. 84, 5094–5097 (2000).
[Crossref] [PubMed]

1999 (2)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594 (1999).
[Crossref]

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767 (1999).
[Crossref]

1997 (2)

S. E. Harris, “Electromagnetically induced transparency,” Phys. Today 50, 36 (1997).
[Crossref]

F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the hanle effect of the d1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997).
[Crossref]

1996 (1)

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 (1)

K.-J. Boller, A. Imamoglu, and S. E. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593 (1991).
[Crossref] [PubMed]

1990 (1)

S. E. Harris, J. E. Field, and A. Imamoglu, “Nonlinear optical processes using electromagnetically induced transparency,” Phys. Rev. Lett. 64, 1107 (1990).
[Crossref] [PubMed]

Abraham, E. R. I.

T. G. Akin, S. P. Krzyzewski, A. M. Marino, and E. R. I. Abraham, “Electromagnetically induced transparency with laguerre-gaussian modes in ultracold rubidium,” Opt. Commun. 339, 209 (2015).
[Crossref]

Agarwal, G. S.

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Akin, T. G.

T. G. Akin, S. P. Krzyzewski, A. M. Marino, and E. R. I. Abraham, “Electromagnetically induced transparency with laguerre-gaussian modes in ultracold rubidium,” Opt. Commun. 339, 209 (2015).
[Crossref]

Akulshin, A. M.

Allen, L.

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

Andre, A.

A. Andre, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin, “Quantum control of light using electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, S589–S604 (2005).
[Crossref]

M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837 (2005).
[Crossref] [PubMed]

Arimondo, E.

F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the hanle effect of the d1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997).
[Crossref]

Bajcsy, M.

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638 (2003).
[Crossref] [PubMed]

Barnett, S. M.

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
[Crossref] [PubMed]

Behroozi, C. H.

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F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the hanle effect of the d1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997).
[Crossref]

Mair, A.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783 (2001).
[Crossref] [PubMed]

Malik, M.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 kilometers,” Proc. Natl Acad. Sci. 113, 13648 (2016).
[Crossref]

Marangos, J. P.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: optics in coherent media,” Rev. Mod. Phys. 77, 633 (2005).
[Crossref]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, “Electromagnetically induced transparency: Optics in coherent media,” Rev. Mod. Phys. 77, 633–673 (2005).
[Crossref]

Marino, A. M.

T. G. Akin, S. P. Krzyzewski, A. M. Marino, and E. R. I. Abraham, “Electromagnetically induced transparency with laguerre-gaussian modes in ultracold rubidium,” Opt. Commun. 339, 209 (2015).
[Crossref]

Massou, F.

M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837 (2005).
[Crossref] [PubMed]

Matsukevich, D. N.

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature 438, 833 (2005).
[Crossref] [PubMed]

Maxein, D.

McGloin, D.

D. McGloin, “Coherent effects in a driven vee scheme,” J. Phys. B: At. Mol. Opt. Phys. 36, 2861 (2003).
[Crossref]

D. McGloin, D. J. Fulton, and M. H. Dunn, “Electromagnetically induced transparency in n-level cascade schemes,” Opt. Commun. 190, 221 (2001).
[Crossref]

McLean, R. J.

Mei, Z.

Y. Hong, Y. Dong, Z. Mei, F. Bo, Z. Yan, and W. Jin-Hui, “Absorption and dispersion control in a five-level m-type atomic system,” Chin. Phys. B 21, 114207 (2012).
[Crossref]

Mekys, A.

J. Ruseckas, A. Mekys, and G. Juzeliūnas, “Slow polaritons with orbital angular momentum in atomic gases,” Phys. Rev. A 83, 023812 (2011).
[Crossref]

Menon, S.

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

Mikhailov, E. E.

Natarajan, V.

S. R. Chanu and V. Natarajan, “Narrowing of resonances in electromagnetically induced transparency and absorption using a laguerre-gaussian control beam,” Opt. Commun. 295, 150 (2013).
[Crossref]

Nicolas, A.

Niu, Y.

X. Yang, K. Ying, Y. Niu, and S. Gong, “Reversible self-kerr nonlinearity in an n-type atomic system through a switching field,” J. Opt. 17, 045505 (2015).
[Crossref]

Novikova, I.

Otterbach, J.

R. G. Unanyan, J. Otterbach, M. Fleischhauer, J. Ruseckas, V. Kudriašov, and G. Juzeliūnas, “Spinor slow-light and dirac particles with variable mass,” Phys. Rev. Lett. 105, 173603 (2010).
[Crossref]

Pack, M. V.

M. V. Pack, R. M. Camacho, and J. C. Howell, “Transients of the electromagnetically-induced-transparency-enhanced refractive kerr nonlinearity: Theory,” Phys. Rev. A 74, 013812 (2006).
[Crossref]

Padgett, M. J.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161 (2011).
[Crossref]

Paspalakis, E.

E. Paspalakis and P. L. Knight, “Electromagnetically induced transparency and controlled group velocity in a multilevel system,” Phys. Rev. A 66, 015802 (2002).
[Crossref]

E. Paspalakis and P. L. Knight, “Transparency, slow light and enhanced nonlinear optics in a four-level scheme,” J. Opt. B: Quantum Semiclass. Opt. 4, S372 (2002).
[Crossref]

Peng, X.

Phillips, D. F.

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783 (2001).
[Crossref] [PubMed]

Piccirillo, B.

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
[Crossref] [PubMed]

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]

Radwell, N.

N. Radwell, T. W. Clark, B. Piccirillo, S. M. Barnett, and S. Franke-Arnold, “Spatially dependent electromagnetically induced transparency,” Phys. Rev. Lett. 114, 123603 (2015).
[Crossref] [PubMed]

Ray, B.

D. Bhattacharyya, B. Ray, and P. N. Ghosh, “Theoretical study of electromagnetically induced transparency in a five-level atom and application to doppler-broadened and doppler-free rb atoms,” J. Phys. B: At. Mol. Opt. Phys. 40, 4061 (2007).
[Crossref]

Renzoni, F.

F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the hanle effect of the d1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997).
[Crossref]

Rochester, S. M.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767 (1999).
[Crossref]

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]

Rostovtsev, Y. V.

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

Ruseckas, J.

H. R. Hamedi, J. Ruseckas, and G. Juzeliūnas, “Electromagnetically induced transparency and nonlinear pulse propagation in a combined tripod and lambda atom-light coupling scheme,” J. Phys. B: At. Mol. Opt. Phys. 50, 185401 (2017).
[Crossref]

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriasov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

J. Ruseckas, V. Kudriašov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
[Crossref]

J. Ruseckas, A. Mekys, and G. Juzeliūnas, “Slow polaritons with orbital angular momentum in atomic gases,” Phys. Rev. A 83, 023812 (2011).
[Crossref]

R. G. Unanyan, J. Otterbach, M. Fleischhauer, J. Ruseckas, V. Kudriašov, and G. Juzeliūnas, “Spinor slow-light and dirac particles with variable mass,” Phys. Rev. Lett. 105, 173603 (2010).
[Crossref]

Sautenkov, V. A.

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

Schmidt, H.

Scully, M. O.

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

Sheng, J.

Sheremet, A. S.

Shi, B.-S.

S. Shi, D.-S. Ding, W. Zhang, Z.-Y. Zhou, M.-X. Dong, S.-L. Liu, K. Wang, B.-S. Shi, and G.-C. Guo, “Transverse azimuthal dephasing of a vortex spin wave in a hot atomic gas,” Phys. Rev. A 95, 033823 (2017).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Linear up-conversion of orbital angular momentum,” Opt. Lett. 37, 3270–3272 (2012).
[Crossref] [PubMed]

Shi, S.

S. Shi, D.-S. Ding, W. Zhang, Z.-Y. Zhou, M.-X. Dong, S.-L. Liu, K. Wang, B.-S. Shi, and G.-C. Guo, “Transverse azimuthal dephasing of a vortex spin wave in a hot atomic gas,” Phys. Rev. A 95, 033823 (2017).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[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]

Slattery, O.

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19, 043001 (2017).
[Crossref] [PubMed]

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]

Tang, X.

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19, 043001 (2017).
[Crossref] [PubMed]

Tu, X. H.

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

Unanyan, R. G.

R. G. Unanyan, J. Otterbach, M. Fleischhauer, J. Ruseckas, V. Kudriašov, and G. Juzeliūnas, “Spinor slow-light and dirac particles with variable mass,” Phys. Rev. Lett. 105, 173603 (2010).
[Crossref]

Ursin, R.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 kilometers,” Proc. Natl Acad. Sci. 113, 13648 (2016).
[Crossref]

Veissier, L.

Walsworth, R. L.

A. Andre, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin, “Quantum control of light using electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, S589–S604 (2005).
[Crossref]

D. F. Phillips, A. Fleischhauer, A. Mair, R. L. Walsworth, and M. D. Lukin, “Storage of light in atomic vapor,” Phys. Rev. Lett. 86, 783 (2001).
[Crossref] [PubMed]

Wang, J.

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

Wang, K.

S. Shi, D.-S. Ding, W. Zhang, Z.-Y. Zhou, M.-X. Dong, S.-L. Liu, K. Wang, B.-S. Shi, and G.-C. Guo, “Transverse azimuthal dephasing of a vortex spin wave in a hot atomic gas,” Phys. Rev. A 95, 033823 (2017).
[Crossref]

Wang, X.-S.

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

Wei, D.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, “Identifying the orbital angular momentum of light based on atomic ensembles,” Eur. Lett. 99, 34003 (2012).
[Crossref]

Welch, G. R.

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

Windholz, L.

F. Renzoni, W. Maichen, L. Windholz, and E. Arimondo, “Coherent population trapping with losses observed on the hanle effect of the d1 sodium line,” Phys. Rev. A 55, 3710–3718 (1997).
[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 (1992).
[Crossref] [PubMed]

Xiang, G.-Y.

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

Xiao, F.

Xiao, M.

Xiong, H. W.

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

Yan, Z.

Y. Hong, Y. Dong, Z. Mei, F. Bo, Z. Yan, and W. Jin-Hui, “Absorption and dispersion control in a five-level m-type atomic system,” Chin. Phys. B 21, 114207 (2012).
[Crossref]

Yang, X.

X. Yang, K. Ying, Y. Niu, and S. Gong, “Reversible self-kerr nonlinearity in an n-type atomic system through a switching field,” J. Opt. 17, 045505 (2015).
[Crossref]

J. Sheng, X. Yang, U. Khadka, and M. Xiao, “All-optical switching in an n-type four-level atom-cavity system,” Opt. Express 19, 17059–17064 (2011).
[Crossref] [PubMed]

Yao, A. M.

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161 (2011).
[Crossref]

Yashchuk, V. V.

D. Budker, D. F. Kimball, S. M. Rochester, and V. V. Yashchuk, “Nonlinear magneto-optics and reduced group velocity of light in atomic vapor with slow ground state relaxation,” Phys. Rev. Lett. 83, 1767 (1999).
[Crossref]

Ying, K.

X. Yang, K. Ying, Y. Niu, and S. Gong, “Reversible self-kerr nonlinearity in an n-type atomic system through a switching field,” J. Opt. 17, 045505 (2015).
[Crossref]

Yu, I. A.

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriasov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

J. Ruseckas, V. Kudriašov, I. A. Yu, and G. Juzeliūnas, “Transfer of orbital angular momentum of light using two-component slow light,” Phys. Rev. A 87, 053840 (2013).
[Crossref]

Zeilinger, A.

M. Krenn, J. Handsteiner, M. Fink, R. Fickler, R. Ursin, M. Malik, and A. Zeilinger, “Twisted light transmission over 143 kilometers,” Proc. Natl Acad. Sci. 113, 13648 (2016).
[Crossref]

R. Fickler, G. Campbell, B. Buchler, P. K. Lam, and A. Zeilinger, “Quantum entanglement of angular momentum states with quantum numbers up to 10010,” Proc. Natl Acad. Sci. 113, 13642 (2016).
[Crossref]

Zhan, M. S.

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

Zhang, P.

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, “Identifying the orbital angular momentum of light based on atomic ensembles,” Eur. Lett. 99, 34003 (2012).
[Crossref]

Zhang, W.

S. Shi, D.-S. Ding, W. Zhang, Z.-Y. Zhou, M.-X. Dong, S.-L. Liu, K. Wang, B.-S. Shi, and G.-C. Guo, “Transverse azimuthal dephasing of a vortex spin wave in a hot atomic gas,” Phys. Rev. A 95, 033823 (2017).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

Zhou, Z.-Y.

S. Shi, D.-S. Ding, W. Zhang, Z.-Y. Zhou, M.-X. Dong, S.-L. Liu, K. Wang, B.-S. Shi, and G.-C. Guo, “Transverse azimuthal dephasing of a vortex spin wave in a hot atomic gas,” Phys. Rev. A 95, 033823 (2017).
[Crossref]

D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, G.-Y. Xiang, X.-S. Wang, Y.-K. Jiang, B.-S. Shi, and G.-C. Guo, “Quantum storage of orbital angular momentum entanglement in an atomic ensemble,” Phys. Rev. Lett. 114, 050502 (2015).
[Crossref] [PubMed]

D.-S. Ding, Z.-Y. Zhou, B.-S. Shi, X.-B. Zou, and G.-C. Guo, “Linear up-conversion of orbital angular momentum,” Opt. Lett. 37, 3270–3272 (2012).
[Crossref] [PubMed]

Zhu, Y.

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

Zibrov, A. S.

M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837 (2005).
[Crossref] [PubMed]

A. Andre, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin, “Quantum control of light using electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, S589–S604 (2005).
[Crossref]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638 (2003).
[Crossref] [PubMed]

Zou, X.-B.

Adv. Opt. Photonics (1)

A. M. Yao and M. J. Padgett, “Orbital angular momentum: origins, behavior and applications,” Adv. Opt. Photonics 3, 161 (2011).
[Crossref]

Chin. Phys. B (1)

Y. Hong, Y. Dong, Z. Mei, F. Bo, Z. Yan, and W. Jin-Hui, “Absorption and dispersion control in a five-level m-type atomic system,” Chin. Phys. B 21, 114207 (2012).
[Crossref]

Eur. Lett. (1)

L. Han, M. Cao, R. Liu, H. Liu, W. Guo, D. Wei, S. Gao, P. Zhang, H. Gao, and F. Li, “Identifying the orbital angular momentum of light based on atomic ensembles,” Eur. Lett. 99, 34003 (2012).
[Crossref]

J. Opt. (2)

X. Yang, K. Ying, Y. Niu, and S. Gong, “Reversible self-kerr nonlinearity in an n-type atomic system through a switching field,” J. Opt. 17, 045505 (2015).
[Crossref]

L. Ma, O. Slattery, and X. Tang, “Optical quantum memory based on electromagnetically induced transparency,” J. Opt. 19, 043001 (2017).
[Crossref] [PubMed]

J. Opt. B: Quantum Semiclass. Opt. (1)

E. Paspalakis and P. L. Knight, “Transparency, slow light and enhanced nonlinear optics in a four-level scheme,” J. Opt. B: Quantum Semiclass. Opt. 4, S372 (2002).
[Crossref]

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

J. Phys. B: At. Mol. Opt. Phys. (4)

D. McGloin, “Coherent effects in a driven vee scheme,” J. Phys. B: At. Mol. Opt. Phys. 36, 2861 (2003).
[Crossref]

D. Bhattacharyya, B. Ray, and P. N. Ghosh, “Theoretical study of electromagnetically induced transparency in a five-level atom and application to doppler-broadened and doppler-free rb atoms,” J. Phys. B: At. Mol. Opt. Phys. 40, 4061 (2007).
[Crossref]

H. R. Hamedi, J. Ruseckas, and G. Juzeliūnas, “Electromagnetically induced transparency and nonlinear pulse propagation in a combined tripod and lambda atom-light coupling scheme,” J. Phys. B: At. Mol. Opt. Phys. 50, 185401 (2017).
[Crossref]

A. Andre, M. D. Eisaman, R. L. Walsworth, A. S. Zibrov, and M. D. Lukin, “Quantum control of light using electromagnetically induced transparency,” J. Phys. B: At. Mol. Opt. Phys. 38, S589–S604 (2005).
[Crossref]

Nat. Commun. (1)

M.-J. Lee, J. Ruseckas, C.-Y. Lee, V. Kudriasov, K.-F. Chang, H.-W. Cho, G. Juzeliūnas, and I. A. Yu, “Experimental demonstration of spinor slow light,” Nat. Commun. 5, 5542 (2014).
[Crossref] [PubMed]

Nature (5)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature 409, 490 (2001).
[Crossref] [PubMed]

M. Bajcsy, A. S. Zibrov, and M. D. Lukin, “Stationary pulses of light in an atomic medium,” Nature 426, 638 (2003).
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L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594 (1999).
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M. D. Eisaman, A. Andre, F. Massou, M. Fleischhauer, A. S. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature 438, 837 (2005).
[Crossref] [PubMed]

T. Chaneliere, D. N. Matsukevich, S. D. Jenkins, S.-Y. Lan, T. A. B. Kennedy, and A. Kuzmich, “Storage and retrieval of single photons transmitted between remote quantum memories,” Nature 438, 833 (2005).
[Crossref] [PubMed]

Opt. Commun. (3)

D. McGloin, D. J. Fulton, and M. H. Dunn, “Electromagnetically induced transparency in n-level cascade schemes,” Opt. Commun. 190, 221 (2001).
[Crossref]

S. R. Chanu and V. Natarajan, “Narrowing of resonances in electromagnetically induced transparency and absorption using a laguerre-gaussian control beam,” Opt. Commun. 295, 150 (2013).
[Crossref]

T. G. Akin, S. P. Krzyzewski, A. M. Marino, and E. R. I. Abraham, “Electromagnetically induced transparency with laguerre-gaussian modes in ultracold rubidium,” Opt. Commun. 339, 209 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

J. Wang, L. B. Kong, X. H. Tu, K. J. Jiang, K. Li, H. W. Xiong, Y. Zhu, and M. S. Zhan, “Electromagnetically induced transparency in multi-level cascade scheme of cold rubidium atoms,” Phys. Lett. A 328, 437 (2004).
[Crossref]

Phys. Rev. A (9)

H. Li, V. A. Sautenkov, Y. V. Rostovtsev, G. R. Welch, P. R. Hemmer, and M. O. Scully, “Electromagnetically induced transparency controlled by a microwave field,” Phys. Rev. A 80, 023820 (2009).
[Crossref]

G. S. Agarwal, T. N. Dey, and S. Menon, “Knob for changing light propagation from subluminal to superluminal,” Phys. Rev. A 64, 053809 (2001).
[Crossref]

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Other (1)

M. Fleischhauer and G. Juzeliūnas, Slow, Stored and Stationary Light(Springer International Publishing, Cham, 2016), pp. 359–383.

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Figures (8)

Fig. 1
Fig. 1 Five-level combined tripod and Λ atomic system (a). Five-level combined tripod and Λ atomic system in the transformed basis for α, β ≠ 0 (b).
Fig. 2
Fig. 2 Different coupling scenarios of CTL atom with structured light, resulting in different regions of spatially structured transparency.
Fig. 3
Fig. 3 Spatially dependent absorption profile of the probe beam in arbitrary units when only the control field Ω2 has an optical vortex. The vorticities are l2 = 1 (a), l2 = 2 (b), l2 = 3 (c), l2 = 4 (d) and l2 = 5 (e). Other parameters are |Ω1| = 0.6Γ,ϵ2 = 0.7Γ, Ω3| = 0.3Γ, |Ω4| = 0.5Γ, Γe = Γb = Γ and Δp = 0.001Γ.
Fig. 4
Fig. 4 Spatially dependent absorption profile of the probe beam in arbitrary units when two control fields Ω2 and Ω3 have optical vortices. The vorticities are l = 1 (a), l = 2 (b), l = 3 (c), l = 4 (d) and l2 = 3, l3 = 4 (e). The remaining parameters are |Ω1| = 0.6Γ, ϵ2 = 0.7Γ, ϵ3 = 0.3Γ, |Ω4| = 0.5Γ, and the other parameters are the same as Fig. 3.
Fig. 5
Fig. 5 Spatially dependent absorption profile of the probe beam in arbitrary units when two control fields Ω3 and Ω4 have optical vortices. The vorticities are l = 1 (a), l = 2 (b), l = 3 (c), l = 4 (d) and l3 = 5, l4 = 6 (e). The selected parameters are |Ω1| = 0.6Γ, Ω2 = 0.7Γ, ϵ3 = 0.3Γ, ϵ4 = 0.5Γ, and the other parameters are the same as Fig. 3.
Fig. 6
Fig. 6 Spatially dependent absorption profile of the probe beam in arbitrary units when two control fields Ω2 and Ω4 have optical vortices. The vorticities are l = 1 (a), l = 2 (b),l = 3 (c), l2 = 4, l4 = −2 (d) and l2 = 5, l4 = 2 (e). The selected parameters are |Ω1| = 0.6Γ, ϵ2 = 0.7Γ, |Ω3| = 0.3Γ, ϵ4 = 0.5Γ, and the other parameters are the same as Fig. 3.
Fig. 7
Fig. 7 Spatially dependent absorption profile of the probe beam in arbitrary units when three control fields Ω2, Ω3 and Ω4 have optical vortices. The vorticities are l = 1 (a), l = 2 (b), l = 3 (c), l2 = 4, l3 = l4 = −2 (d) and l2 = 4, l3 = −2, l4 = −5 (e). The selected parameters are |Ω1| = 0.6Γ, ϵ2 = 0.7Γ, ϵ3 = 0.3Γ, ϵ4 = 0.5Γ, and the other parameters are the same as Fig. 3.
Fig. 8
Fig. 8 Spatially dependent absorption profile of the probe beam in arbitrary units when all control fields are vortex beams. The vorticities are l = 1 (a), l = 2 (b), l = 3 (c), l1 = −2, l2 = 1, l3 = 1, l4 = −1 (d) and l1 = 1, l2 = 2, l3 = 4, l4 = 3 (e). The selected parameters are ϵ1 = 0.6Γ, ϵ2 = 0.7Γ, ϵ3 = 0.3Γ, ϵ4 = 0.5Γ, and the other parameters are the same as Fig. 3.

Equations (54)

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H 4 Levels = Ω 1 * | c b | Ω 2 * | d b | Ω 3 * | c e | Ω 4 * | d e | + H .c ..
H 5 Levels = ( Ω p * | a b | + Ω p | b a | ) + H 4 Levels .
H 4 Levels = β | D e b | α | B e b | Ω | B e e | + H . c . ,
| D e = 1 Ω ( Ω 4 | c Ω 3 | d ) ,
| B e = 1 Ω ( Ω 3 * | c + Ω 4 * | d ) ,
β = 1 Ω ( Ω 1 * Ω 4 * Ω 2 * Ω 3 * ) ,
α = 1 Ω ( Ω 1 * Ω 3 + Ω 2 * Ω 4 ) ,
Ω = | Ω 3 | 2 + | Ω 4 | 2 .
| D = β | a Ω p | D e ,
ρ ˙ b a = ( Γ b / 2 i Δ p ) ρ b a + i α ρ B e a + i β ρ D e a + i Ω p ,
ρ ˙ B e a = i Δ p ρ B e a + i α * ρ b a + i Ω * ρ e a ,
ρ ˙ D e a = i Δ p ρ D e a + i β * ρ b a ,
ρ ˙ e a = ( Γ e / 2 i Δ p ) ρ e a + i Ω ρ B e a ,
ρ b a = Ω p Δ p ( | Ω | 2 + i Δ p ( Γ e / 2 i Δ p ) ) i Δ p ( Γ e / 2 i Δ p ) ζ + i | Ω | 2 Δ p ( Γ b / 2 i Δ p ) + ( Γ b / 2 i Δ p ) Δ p 2 ( Γ e / 2 i Δ p ) | Ω | 2 | β | 2 ,
Im ( ρ b a ) = Ω p | Ω | 2 B Δ p + Δ p 2 A Γ e / 2 Δ p 3 B A 2 + B 2 ,
A = Δ p 4 + Δ p 2 ( ζ + | Ω | 2 + Γ e Γ b / 4 ) | Ω | 2 | β | 2 ,
B = Δ p 3 + ( Γ e + Γ b / 2 + Δ p ( | Ω | 2 Γ b / 2 + ζ Γ e / 2 ) .
| Ω | 2 | β | 2 = | Ω 1 | 2 | Ω 4 | 2 + | Ω 2 | 2 | Ω 3 | 2 Q ,
Q = Ω 1 * Ω 2 Ω 3 Ω 4 * + Ω 1 Ω 2 * Ω 3 * Ω 4 ,
| Ω j | = ϵ j ( r w ) | l j | exp ( r 2 w 2 ) ,
Ω 2 = | Ω 2 | exp ( i l 2 Φ ) ,
Ω 1 = | Ω 1 | ,
Ω 3 = | Ω 3 | ,
Ω 4 = | Ω 4 | .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( l 2 Φ ) .
ρ b a = Ω p ( | Ω | 2 + i Δ p ( Γ e / 2 i Δ p ) ) i ( Γ e / 2 i Δ p ) ζ + i | Ω | 2 ( Γ b / 2 i Δ p ) + ( Γ b / 2 i Δ p ) Δ p ( Γ e / 2 i Δ p ) .
Ω 2 = | Ω 2 | exp ( i l 2 Φ ) ,
Ω 3 = | Ω 3 | exp ( i l 3 Φ ) ,
Ω 1 = | Ω 1 | ,
Ω 4 = | Ω 4 | .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( ( l 2 + l 3 ) Φ ) .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( 2 l Φ ) ,
Ω 3 = | Ω 3 | exp ( i l 3 Φ ) ,
Ω 4 = | Ω 4 | exp ( i l 4 Φ ) ,
Ω 1 = | Ω 1 | ,
Ω 2 = | Ω 2 | .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( ( l 3 l 4 ) Φ ) .
Ω 2 = | Ω 2 | exp ( i l 2 Φ ) ,
Ω 4 = | Ω 4 | exp ( i l 4 Φ ) ,
Ω 1 = | Ω 1 | ,
Ω 3 = | Ω 3 | .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( ( l 2 l 4 ) Φ ) .
Ω 2 = | Ω 2 | exp ( i l 2 Φ ) ,
Ω 3 = | Ω 3 | exp ( i l 3 Φ ) ,
Ω 4 = | Ω 4 | exp ( i l 4 Φ ) ,
Ω 1 = | Ω 1 | ,
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( ( l 2 + l 3 l 4 ) Φ ) .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( 3 l Φ ) .
Ω 1 = | Ω 1 | exp ( i l 1 Φ ) ,
Ω 2 = | Ω 2 | exp ( i l 2 Φ ) ,
Ω 3 = | Ω 3 | exp ( i l 3 Φ ) ,
Ω 4 = | Ω 4 | exp ( i l 4 Φ ) ,
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( ( l 2 + l 3 l 1 l 4 ) Φ ) .
Q = 2 | Ω 1 | | Ω 2 | | Ω 3 | | Ω 4 | cos ( 4 Φ ) ,

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