Abstract

We have studied electromagnetically induced transparency (EIT) in the 5S125P325D52 ladder-type system of Rb. We observed relative changing magnitude of EIT hyperfine structures depending on not only the polarizations of the lasers but also the intensity of the coupling laser. The coupling-intensity effects are attributed to the nonlinear increase of the EIT signal to the coupling intensity. EIT signals nonlinear on the coupling intensity are analyzed by considering coherent interaction between atom and laser fields.

© 2005 Optical Society of America

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  1. S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
    [CrossRef]
  2. K. J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
    [CrossRef] [PubMed]
  3. M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
    [CrossRef] [PubMed]
  4. H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).
  5. 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, 2053-2056 (1996).
    [CrossRef] [PubMed]
  6. B. S. Ham, S. M. Shahriar, and P. R. Hemmer, "Electromagnetically induced transparency over spectral hole-burning temperature in a rare-earthdoped solid," J. Opt. Soc. Am. B 16, 801-804 (1999).
    [CrossRef]
  7. S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
    [CrossRef]
  8. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
    [CrossRef]
  9. S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
    [CrossRef]
  10. D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
    [CrossRef] [PubMed]
  11. D. McGloin, M. H. Dunn, and D. J. Fulton, "Polarization effects in electromagnetically induced transparency," Phys. Rev. A 62, 053802 (2000).
    [CrossRef]
  12. S. D. Badger, I. G. Hughes, and C. S. Adams, "Hyperfine effects in electromagnetically induced transparency," J. Phys. B 34, L749-L756 (2001).
    [CrossRef]
  13. J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
    [CrossRef] [PubMed]
  14. B. Lü, W. H. Burkett, and M. Xiao, "Electromagnetically induced transparency with variable coupling-laser linewidth," Phys. Rev. A 56, 976-979 (1997).
    [CrossRef]
  15. M. Stähler, R. Wynands, S. Knappe, J. Kitching, L. Hollberg, A. Taichenachev, and V. Yudin, "Coherent population trapping resonances in thermal Rb85 vapor: D1 versus D2 line excitation," Opt. Lett. 27, 1472-1474 (2002).
    [CrossRef]

2002 (1)

2001 (1)

S. D. Badger, I. G. Hughes, and C. S. Adams, "Hyperfine effects in electromagnetically induced transparency," J. Phys. B 34, L749-L756 (2001).
[CrossRef]

2000 (1)

D. McGloin, M. H. Dunn, and D. J. Fulton, "Polarization effects in electromagnetically induced transparency," Phys. Rev. A 62, 053802 (2000).
[CrossRef]

1999 (4)

H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).

B. S. Ham, S. M. Shahriar, and P. R. Hemmer, "Electromagnetically induced transparency over spectral hole-burning temperature in a rare-earthdoped solid," J. Opt. Soc. Am. B 16, 801-804 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

1997 (3)

S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
[CrossRef]

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

B. Lü, W. H. Burkett, and M. Xiao, "Electromagnetically induced transparency with variable coupling-laser linewidth," Phys. Rev. A 56, 976-979 (1997).
[CrossRef]

1996 (1)

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, 2053-2056 (1996).
[CrossRef] [PubMed]

1995 (3)

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

1991 (1)

K. J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Adams, C. S.

S. D. Badger, I. G. Hughes, and C. S. Adams, "Hyperfine effects in electromagnetically induced transparency," J. Phys. B 34, L749-L756 (2001).
[CrossRef]

Badger, S. D.

S. D. Badger, I. G. Hughes, and C. S. Adams, "Hyperfine effects in electromagnetically induced transparency," J. Phys. B 34, L749-L756 (2001).
[CrossRef]

Behroozi, C. H.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Boller, K. J.

K. J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Burkett, W. H.

B. Lü, W. H. Burkett, and M. Xiao, "Electromagnetically induced transparency with variable coupling-laser linewidth," Phys. Rev. A 56, 976-979 (1997).
[CrossRef]

Chen, H. X.

S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
[CrossRef]

Dunn, M. H.

D. McGloin, M. H. Dunn, and D. J. Fulton, "Polarization effects in electromagnetically induced transparency," Phys. Rev. A 62, 053802 (2000).
[CrossRef]

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

Durrant, A. V.

S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
[CrossRef]

Dutton, Z.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

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, 2053-2056 (1996).
[CrossRef] [PubMed]

Fulton, D. J.

D. McGloin, M. H. Dunn, and D. J. Fulton, "Polarization effects in electromagnetically induced transparency," Phys. Rev. A 62, 053802 (2000).
[CrossRef]

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

Gea-Banacloche, J.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Ham, B. S.

Harris, S. E.

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

K. J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Hau, L. V.

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Hemmer, P. R.

Hollberg, L.

Hopkins, S. A.

S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
[CrossRef]

Hughes, I. G.

S. D. Badger, I. G. Hughes, and C. S. Adams, "Hyperfine effects in electromagnetically induced transparency," J. Phys. B 34, L749-L756 (2001).
[CrossRef]

Imamoglu, A.

K. J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

Jin, S.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Jin, S.-Z.

J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Kim, B. S.

H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).

Kim, H. A.

H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).

Kim, J. B.

H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).

Kitching, J.

Knappe, S.

Li, Y.

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Li, Y.-Q.

J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

Lü, B.

B. Lü, W. H. Burkett, and M. Xiao, "Electromagnetically induced transparency with variable coupling-laser linewidth," Phys. Rev. A 56, 976-979 (1997).
[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, 2053-2056 (1996).
[CrossRef] [PubMed]

McGloin, D.

D. McGloin, M. H. Dunn, and D. J. Fulton, "Polarization effects in electromagnetically induced transparency," Phys. Rev. A 62, 053802 (2000).
[CrossRef]

Moon, H. S.

H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).

Moseley, R. R.

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

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, 2053-2056 (1996).
[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, 2053-2056 (1996).
[CrossRef] [PubMed]

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, 2053-2056 (1996).
[CrossRef] [PubMed]

Shahriar, S. M.

Shepherd, S.

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

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, 2053-2056 (1996).
[CrossRef] [PubMed]

Sinclair, B. D.

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

Stähler, M.

Taichenachev, A.

Usadi, E.

S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
[CrossRef]

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, 2053-2056 (1996).
[CrossRef] [PubMed]

Wynands, R.

Xiao, M.

B. Lü, W. H. Burkett, and M. Xiao, "Electromagnetically induced transparency with variable coupling-laser linewidth," Phys. Rev. A 56, 976-979 (1997).
[CrossRef]

J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Yudin, V.

J. Korean Phys. Soc. (1)

H. S. Moon, H. A. Kim, B. S. Kim, and J. B. Kim, "Electromagnetically induced transparency in an ideal three level system in Rb87 atoms," J. Korean Phys. Soc. 35, 207-211 (1999).

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

J. Phys. B (1)

S. D. Badger, I. G. Hughes, and C. S. Adams, "Hyperfine effects in electromagnetically induced transparency," J. Phys. B 34, L749-L756 (2001).
[CrossRef]

Nature (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, "Light speed reduction to 17 metresper second in an ultracold atomic gas," Nature 397, 594-598 (1999).
[CrossRef]

Opt. Commun. (1)

S. A. Hopkins, E. Usadi, H. X. Chen, and A. V. Durrant, "Electromagnetically induced transparency of laser-cooled rubidium atoms in three-level Lambda-type systems," Opt. Commun. 138, 185-192 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (4)

J. Gea-Banacloche, Y.-Q. Li, S.-Z. Jin, and M. Xiao, "Electromagnetically induced transparency in ladder-type inhomogeneously broadened media: theory and experiment," Phys. Rev. A 51, 576-584 (1995).
[CrossRef] [PubMed]

B. Lü, W. H. Burkett, and M. Xiao, "Electromagnetically induced transparency with variable coupling-laser linewidth," Phys. Rev. A 56, 976-979 (1997).
[CrossRef]

D. J. Fulton, S. Shepherd, R. R. Moseley, B. D. Sinclair, and M. H. Dunn, "Continuous-wave electromagnetically induced transparency: a comparison of V, Lambda, and cascade systems," Phys. Rev. A 52, 2302-2311 (1995).
[CrossRef] [PubMed]

D. McGloin, M. H. Dunn, and D. J. Fulton, "Polarization effects in electromagnetically induced transparency," Phys. Rev. A 62, 053802 (2000).
[CrossRef]

Phys. Rev. Lett. (4)

S. E. Harris and L. V. Hau, "Nonlinear optics at low light levels," Phys. Rev. Lett. 82, 4611-4614 (1999).
[CrossRef]

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, 2053-2056 (1996).
[CrossRef] [PubMed]

K. J. Boller, A. Imamoglu, and S. E. Harris, "Observation of electromagnetically induced transparency," Phys. Rev. Lett. 66, 2593-2596 (1991).
[CrossRef] [PubMed]

M. Xiao, Y. Li, S. Jin, and J. Gea-Banacloche, "Measurement of dispersive properties of electromagnetically induced transparency in rubidium atoms," Phys. Rev. Lett. 74, 666-669 (1995).
[CrossRef] [PubMed]

Phys. Today (1)

S. E. Harris, "Electromagnetically induced transparency," Phys. Today 50, 36-42 (1997).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Energy-level diagram of the 5 S 1 2 5 P 3 2 5 D 5 2 transition of an Rb atom.

Fig. 2
Fig. 2

Experimental setup for observation of a ladder-type EIT spectrum. BS, beam splitter; AP, aperture; PD, photo diode; QWP, quarter-wave plate; and HWP, half-wave plate.

Fig. 3
Fig. 3

Ladder-type EIT spectrum of the 5 S 1 2 5 P 3 2 5 D 5 2 transition. Black curve, EIT spectrum; gray curve, the saturated absorption spectrum of the probe laser.

Fig. 4
Fig. 4

(a) EIT spectra according to the polarizations of the lasers in the 5 S 1 2 ( F = 2 ) 5 P 3 2 ( F = 1 , 2 , 3 ) 5 D 5 2 ( F = 2 , 3 , 4 ) transition of Rb 87 . (b) Calculated spectra using the two-photon transition probability.

Fig. 5
Fig. 5

EIT spectra according to the coupling intensity in the 5 S 1 2 ( F = 2 ) 5 P 3 2 ( F = 1 , 2 , 3 ) 5 D 5 2 ( F = 2 , 3 , 4 ) transition of Rb 87 .

Fig. 6
Fig. 6

(a) Magnitude of EIT signals in the 5 D 5 2 ( F = 3 ) and the 5 D 5 2 ( F = 4 ) transitions of Rb 87 . (b) Relative absorption reduction in the 5 D 5 2 ( F = 4 ) transition to one in the 5 D 5 2 ( F = 3 ) transition.

Fig. 7
Fig. 7

(a) In the cases of a 2 = 0.2 and a 2 = 1.0 , the calculated intensity-absorption coefficient as a function of the coupling intensity. (b) Calculated ratio of two intensity-absorption coefficients, α ( 1.0 ) α ( 0.2 ) .

Equations (3)

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χ = 4 i c g 12 2 N 0 π ϵ 0 u ω p e z 2 [ 1 erf ( z ) ] ,
z = c u ω p [ γ 21 i Δ 1 + Ω c 2 4 γ 31 i ( Δ 1 + Δ 2 ) ] ,
z r = c u ω p ( γ 21 + a 2 R γ 31 I ) ,

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