Abstract

We propose a new scheme for achieving giant enhancement of the self-Kerr nonlinearity in a four-level Y-type atom with two highest nearly degenerate lying levels. We have found that, owing to the quantum interference effect that originates from spontaneous emission from two closely lying levels, large Kerr nonlinearity can be achieved under appropriate conditions. In particular, in addition to the amplitudes, the phases of the applied fields also affect the Kerr nonlinearity. We attribute the enhancement of Kerr nonlinearity mainly to the presence of some extra atomic coherence terms induced by the spontaneously generated coherence. We present a physical understanding of our numerical results using analytical explanation.

© 2009 Optical Society of America

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  1. J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407-412 (1992).
    [CrossRef]
  2. P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77, 3995-3998 (1996).
    [CrossRef] [PubMed]
  3. R. M. Whitley and C. R. Stroud, Jr., “Double optical resonance,” Phys. Rev. A 14, 1498-1512 (1976).
    [CrossRef]
  4. Z. Ficek, B. J. Dalton, and P. L. Knight, “Fluorescence intensity and squeezing in 3-level atom-ladder case,” Phys. Rev. A 51, 4062-4065 (1995).
    [CrossRef] [PubMed]
  5. H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
    [CrossRef]
  6. W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
    [CrossRef]
  7. S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61, 013807 (1999).
    [CrossRef]
  8. S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).
  9. J. Evers, D. Bullock, and C. H. Keitel, “Dark state suppression and narrow fluorescent feature in a laser-driven Λ atom,” Opt. Commun. 209, 173-179 (2002).
    [CrossRef]
  10. E. Paspalakis, S. Q. Gong, and P. L. Knight, “Spontaneous emission-induced coherence effects in absorption and dispersion of a V-type three-level atom,” Opt. Commun. 152, 293-297 (1998).
    [CrossRef]
  11. L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
    [CrossRef] [PubMed]
  12. J. Wang, H. M. Wiseman, and Z. Ficek, “Quantum interference in the fluorescenceof a molecular system,” Phys. Rev. A 62, 013818 (2000).
    [CrossRef]
  13. P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
    [CrossRef]
  14. A. Joshi, W. G. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203-207 (2003).
    [CrossRef]
  15. C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
    [CrossRef]
  16. H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
    [CrossRef]
  17. S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76, 388-391 (1996).
    [CrossRef] [PubMed]
  18. A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
    [CrossRef] [PubMed]
  19. E. Arimondo, “Coherent population trapping in laser spectroscopy,” Progress in Optics, E.Wolf, ed. (Amsterdam, 1996), pp. 257-354.
    [CrossRef]
  20. J. H. Wu and J. Y. Gao, “Phase control of the probe gain without population inversion in a four-level V model,” J. Opt. Soc. Am. B 19, 2863-2866 (2002).
    [CrossRef]
  21. C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
    [CrossRef]
  22. H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
    [CrossRef]
  23. M. A. Antón, O. G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
    [CrossRef]
  24. B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
    [CrossRef]
  25. H. R. Xia, C. Y. Ye, and S. Y. Zhu, “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 77, 1032-1034 (1996).
    [CrossRef] [PubMed]
  26. P. R. Berman, “Analysis of dynamical suppression of spontaneous emission,” Phys. Rev. A 58, 4886-4891 (1998).
    [CrossRef]
  27. M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
    [CrossRef]
  28. Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
    [CrossRef]
  29. A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
    [CrossRef]
  30. S. E. Harris and L. V. Hau, “Nonlinear optics at low light levels,” Phys. Rev. Lett. 82, 4611-4614 (1999).
    [CrossRef]
  31. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
    [CrossRef]
  32. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
    [CrossRef] [PubMed]
  33. H. Kang and Y. F. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
    [CrossRef] [PubMed]
  34. R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77, 033820 (2008).
    [CrossRef]
  35. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Univ. Press, 1997).
  36. G. S. Agarwal, Quantum Optics (Springer-Verlag, 1974).
  37. Y. P. Niu and S. Q. Gong, “Enhancing Kerr nonlinearity via spontaneously generated coherence,” Phys. Rev. A 73, 053811(6) (2006).
    [CrossRef]

2008 (2)

H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
[CrossRef]

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77, 033820 (2008).
[CrossRef]

2006 (1)

Y. P. Niu and S. Q. Gong, “Enhancing Kerr nonlinearity via spontaneously generated coherence,” Phys. Rev. A 73, 053811(6) (2006).
[CrossRef]

2005 (3)

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

M. A. Antón, O. G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

2004 (3)

B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
[CrossRef]

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
[CrossRef]

2003 (3)

A. Joshi, W. G. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203-207 (2003).
[CrossRef]

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

H. Kang and Y. F. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

2002 (5)

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef] [PubMed]

J. H. Wu and J. Y. Gao, “Phase control of the probe gain without population inversion in a four-level V model,” J. Opt. Soc. Am. B 19, 2863-2866 (2002).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

J. Evers, D. Bullock, and C. H. Keitel, “Dark state suppression and narrow fluorescent feature in a laser-driven Λ atom,” Opt. Commun. 209, 173-179 (2002).
[CrossRef]

2000 (2)

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

J. Wang, H. M. Wiseman, and Z. Ficek, “Quantum interference in the fluorescenceof a molecular system,” Phys. Rev. A 62, 013818 (2000).
[CrossRef]

1999 (3)

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61, 013807 (1999).
[CrossRef]

S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).

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

1998 (3)

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

P. R. Berman, “Analysis of dynamical suppression of spontaneous emission,” Phys. Rev. A 58, 4886-4891 (1998).
[CrossRef]

E. Paspalakis, S. Q. Gong, and P. L. Knight, “Spontaneous emission-induced coherence effects in absorption and dispersion of a V-type three-level atom,” Opt. Commun. 152, 293-297 (1998).
[CrossRef]

1997 (2)

H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
[CrossRef]

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

1996 (3)

H. R. Xia, C. Y. Ye, and S. Y. Zhu, “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 77, 1032-1034 (1996).
[CrossRef] [PubMed]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77, 3995-3998 (1996).
[CrossRef] [PubMed]

1995 (2)

Z. Ficek, B. J. Dalton, and P. L. Knight, “Fluorescence intensity and squeezing in 3-level atom-ladder case,” Phys. Rev. A 51, 4062-4065 (1995).
[CrossRef] [PubMed]

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

1992 (1)

J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407-412 (1992).
[CrossRef]

1976 (1)

R. M. Whitley and C. R. Stroud, Jr., “Double optical resonance,” Phys. Rev. A 14, 1498-1512 (1976).
[CrossRef]

Agarwal, G. S.

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61, 013807 (1999).
[CrossRef]

G. S. Agarwal, Quantum Optics (Springer-Verlag, 1974).

Antón, M. A.

M. A. Antón, O. G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

Arimondo, E.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Progress in Optics, E.Wolf, ed. (Amsterdam, 1996), pp. 257-354.
[CrossRef]

Arun, R.

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77, 033820 (2008).
[CrossRef]

Berman, P. R.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

P. R. Berman, “Analysis of dynamical suppression of spontaneous emission,” Phys. Rev. A 58, 4886-4891 (1998).
[CrossRef]

Bracker, A. S.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Bullock, D.

J. Evers, D. Bullock, and C. H. Keitel, “Dark state suppression and narrow fluorescent feature in a laser-driven Λ atom,” Opt. Commun. 209, 173-179 (2002).
[CrossRef]

Calderón, O. G.

M. A. Antón, O. G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

Carreño, F.

M. A. Antón, O. G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

Cheng, J.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Dalton, B. J.

Z. Ficek, B. J. Dalton, and P. L. Knight, “Fluorescence intensity and squeezing in 3-level atom-ladder case,” Phys. Rev. A 51, 4062-4065 (1995).
[CrossRef] [PubMed]

Deutsch, M.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

Dong, P.

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

Du, S. D.

S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).

Evers, J.

J. Evers, D. Bullock, and C. H. Keitel, “Dark state suppression and narrow fluorescent feature in a laser-driven Λ atom,” Opt. Commun. 209, 173-179 (2002).
[CrossRef]

Fan, X. J.

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
[CrossRef]

Ficek, Z.

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

J. Wang, H. M. Wiseman, and Z. Ficek, “Quantum interference in the fluorescenceof a molecular system,” Phys. Rev. A 62, 013818 (2000).
[CrossRef]

Z. Ficek, B. J. Dalton, and P. L. Knight, “Fluorescence intensity and squeezing in 3-level atom-ladder case,” Phys. Rev. A 51, 4062-4065 (1995).
[CrossRef] [PubMed]

Gao, J. Y.

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

J. H. Wu and J. Y. Gao, “Phase control of the probe gain without population inversion in a four-level V model,” J. Opt. Soc. Am. B 19, 2863-2866 (2002).
[CrossRef]

Gong, S. Q.

H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
[CrossRef]

Y. P. Niu and S. Q. Gong, “Enhancing Kerr nonlinearity via spontaneously generated coherence,” Phys. Rev. A 73, 053811(6) (2006).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
[CrossRef]

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).

E. Paspalakis, S. Q. Gong, and P. L. Knight, “Spontaneous emission-induced coherence effects in absorption and dispersion of a V-type three-level atom,” Opt. Commun. 152, 293-297 (1998).
[CrossRef]

Gurudev, M. V.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Harris, S. E.

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

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

Hau, L. V.

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

Hood, C. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

Hou, B. P.

B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
[CrossRef]

Imamoglu, A.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

Javanainen, J.

J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407-412 (1992).
[CrossRef]

Jin, S. Q.

H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

Joshi, A.

A. Joshi, W. G. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203-207 (2003).
[CrossRef]

Kang, H.

H. Kang and Y. F. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

Keitel, C. H.

J. Evers, D. Bullock, and C. H. Keitel, “Dark state suppression and narrow fluorescent feature in a laser-driven Λ atom,” Opt. Commun. 209, 173-179 (2002).
[CrossRef]

Kimble, H. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

Knight, P. L.

E. Paspalakis, S. Q. Gong, and P. L. Knight, “Spontaneous emission-induced coherence effects in absorption and dispersion of a V-type three-level atom,” Opt. Commun. 152, 293-297 (1998).
[CrossRef]

Z. Ficek, B. J. Dalton, and P. L. Knight, “Fluorescence intensity and squeezing in 3-level atom-ladder case,” Phys. Rev. A 51, 4062-4065 (1995).
[CrossRef] [PubMed]

Lange, W.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

Lazarov, G.

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

Lee, H.

H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
[CrossRef]

Li, B.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Li, L.

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

Li, X. Q.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Li, Y.

S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).

Litvak, A. G.

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef] [PubMed]

Liu, C. P.

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
[CrossRef]

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

Lyyra, A. M.

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

Ma, H. M.

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

Mabuchi, H.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

Menon, S.

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61, 013807 (1999).
[CrossRef]

Niu, Y. P.

H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
[CrossRef]

Y. P. Niu and S. Q. Gong, “Enhancing Kerr nonlinearity via spontaneously generated coherence,” Phys. Rev. A 73, 053811(6) (2006).
[CrossRef]

Paspalakis, E.

E. Paspalakis, S. Q. Gong, and P. L. Knight, “Spontaneous emission-induced coherence effects in absorption and dispersion of a V-type three-level atom,” Opt. Commun. 152, 293-297 (1998).
[CrossRef]

Polynkin, P.

H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
[CrossRef]

Qi, J.

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

Schmidt, H.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

Scully, M. O.

H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
[CrossRef]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Univ. Press, 1997).

Sham, L. J.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Steel, D. G.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Stroud, C. R.

R. M. Whitley and C. R. Stroud, Jr., “Double optical resonance,” Phys. Rev. A 14, 1498-1512 (1976).
[CrossRef]

Sun, H.

H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
[CrossRef]

Sun, W. L.

B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
[CrossRef]

Sun, Z.

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

Swain, S.

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77, 3995-3998 (1996).
[CrossRef] [PubMed]

Tang, S. H.

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

Tokman, M. D.

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef] [PubMed]

Turchette, Q. A.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

Wang, J.

J. Wang, H. M. Wiseman, and Z. Ficek, “Quantum interference in the fluorescenceof a molecular system,” Phys. Rev. A 62, 013818 (2000).
[CrossRef]

Wang, S. J.

B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
[CrossRef]

Wang, X.

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

Whitley, R. M.

R. M. Whitley and C. R. Stroud, Jr., “Double optical resonance,” Phys. Rev. A 14, 1498-1512 (1976).
[CrossRef]

Wiseman, H. M.

J. Wang, H. M. Wiseman, and Z. Ficek, “Quantum interference in the fluorescenceof a molecular system,” Phys. Rev. A 62, 013818 (2000).
[CrossRef]

Woods, G.

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

Wu, J. H.

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

J. H. Wu and J. Y. Gao, “Phase control of the probe gain without population inversion in a four-level V model,” J. Opt. Soc. Am. B 19, 2863-2866 (2002).
[CrossRef]

Xia, H. R.

H. R. Xia, C. Y. Ye, and S. Y. Zhu, “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 77, 1032-1034 (1996).
[CrossRef] [PubMed]

Xiao, M.

A. Joshi, W. G. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203-207 (2003).
[CrossRef]

Xu, W. H.

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

Xu, X. D.

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

Xu, Z.

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

Xu, Z. Z.

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
[CrossRef]

S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).

Yamamoto, Y.

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

Yang, J.

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

Yang, W. G.

A. Joshi, W. G. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203-207 (2003).
[CrossRef]

Ye, C. Y.

H. R. Xia, C. Y. Ye, and S. Y. Zhu, “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 77, 1032-1034 (1996).
[CrossRef] [PubMed]

Yu, W. L.

B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
[CrossRef]

Zhou, P.

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77, 3995-3998 (1996).
[CrossRef] [PubMed]

Zhu, S. Y.

H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
[CrossRef]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

H. R. Xia, C. Y. Ye, and S. Y. Zhu, “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 77, 1032-1034 (1996).
[CrossRef] [PubMed]

Zhu, Y. F.

H. Kang and Y. F. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Univ. Press, 1997).

Europhys. Lett. (1)

J. Javanainen, “Effect of state superpositions created by spontaneous emission on laser-driven transitions,” Europhys. Lett. 17, 407-412 (1992).
[CrossRef]

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

J. Phys. B (1)

H. Sun, Y. P. Niu, S. Q. Jin, and S. Q. Gong, “Phase control of the Kerr nonlinearity in electromagnetically induced transparency media,” J. Phys. B 41, 065504 (2008).
[CrossRef]

Opt. Commun. (5)

C. P. Liu, S. Q. Gong, X. J. Fan, S. Q. Jin, and Z. Z. Xu, “Effect of phase on the dynamic evolution of a Λ system with spontaneously generated coherence,” Opt. Commun. 254, 368-345 (2005).
[CrossRef]

H. M. Ma, S. Q. Gong, C. P. Liu, Z. Sun, and Z. Xu, “Effects of spontaneous emission-induced coherence on population inversion in a ladder-type atomic system,” Opt. Commun. 223, 97-101 (2003).
[CrossRef]

J. Evers, D. Bullock, and C. H. Keitel, “Dark state suppression and narrow fluorescent feature in a laser-driven Λ atom,” Opt. Commun. 209, 173-179 (2002).
[CrossRef]

E. Paspalakis, S. Q. Gong, and P. L. Knight, “Spontaneous emission-induced coherence effects in absorption and dispersion of a V-type three-level atom,” Opt. Commun. 152, 293-297 (1998).
[CrossRef]

C. P. Liu, S. Q. Gong, X. J. Fan, and Z. Z. Xu, “Phase control of spontaneously generated coherence induced bistability,” Opt. Commun. 239, 383-389 (2004).
[CrossRef]

Phys. Lett. A (1)

A. Joshi, W. G. Yang, and M. Xiao, “Effect of spontaneously generated coherence on optical bistability in three-level Λ-type atomic system,” Phys. Lett. A 315, 203-207 (2003).
[CrossRef]

Phys. Rev. A (14)

Y. P. Niu and S. Q. Gong, “Enhancing Kerr nonlinearity via spontaneously generated coherence,” Phys. Rev. A 73, 053811(6) (2006).
[CrossRef]

R. Arun, “Interference-induced splitting of resonances in spontaneous emission,” Phys. Rev. A 77, 033820 (2008).
[CrossRef]

Z. Ficek and S. Swain, “Simulating quantum interference in a three-level system with perpendicular transition dipole moments,” Phys. Rev. A 69, 023401 (2004).
[CrossRef]

P. R. Berman, “Analysis of dynamical suppression of spontaneous emission,” Phys. Rev. A 58, 4886-4891 (1998).
[CrossRef]

M. A. Antón, O. G. Calderón, and F. Carreño, “Spontaneously generated coherence effects in a laser-driven four-level atomic system,” Phys. Rev. A 72, 023809 (2005).
[CrossRef]

B. P. Hou, S. J. Wang, W. L. Yu, and W. L. Sun, “Effect of vacuum-induced coherence on single- and two-photon absorption in a four-level Y-type atomic system,” Phys. Rev. A 69, 053805 (2004).
[CrossRef]

H. Lee, P. Polynkin, M. O. Scully, and S. Y. Zhu, “Quenching of spontaneous emission via quantum interference,” Phys. Rev. A 55, 4454-4465 (1997).
[CrossRef]

J. Wang, H. M. Wiseman, and Z. Ficek, “Quantum interference in the fluorescenceof a molecular system,” Phys. Rev. A 62, 013818 (2000).
[CrossRef]

P. Dong and S. H. Tang, “Absorption spectrum of a V-type three-level atom driven by a coherent field,” Phys. Rev. A 65, 033816 (2002).
[CrossRef]

W. H. Xu, J. H. Wu, and J. Y. Gao, “Effects of spontaneously generated coherence on transient process in a Λ system,” Phys. Rev. A 66, 063812 (2002).
[CrossRef]

S. Menon and G. S. Agarwal, “Gain components in the Autler-Townes doublet from quantum interferences in decay channels,” Phys. Rev. A 61, 013807 (1999).
[CrossRef]

S. Q. Gong, Y. Li, S. D. Du, and Z. Z. Xu, “Unexpected population inversion via spontaneously generated coherence of a lambda system,” Phys. Rev. A 259, 43-48 (1999).

R. M. Whitley and C. R. Stroud, Jr., “Double optical resonance,” Phys. Rev. A 14, 1498-1512 (1976).
[CrossRef]

Z. Ficek, B. J. Dalton, and P. L. Knight, “Fluorescence intensity and squeezing in 3-level atom-ladder case,” Phys. Rev. A 51, 4062-4065 (1995).
[CrossRef] [PubMed]

Phys. Rev. Lett. (11)

P. Zhou and S. Swain, “Ultranarrow spectral lines via quantum interference,” Phys. Rev. Lett. 77, 3995-3998 (1996).
[CrossRef] [PubMed]

L. Li, X. Wang, J. Yang, G. Lazarov, J. Qi, and A. M. Lyyra, “Comment on “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 84, 4016-4016 (2000).
[CrossRef] [PubMed]

S. Y. Zhu and M. O. Scully, “Spectral line elimination and spontaneous emission cancellation via quantum interference,” Phys. Rev. Lett. 76, 388-391 (1996).
[CrossRef] [PubMed]

A. G. Litvak and M. D. Tokman, “Electromagnetically induced transparency in ensembles of classical oscillators,” Phys. Rev. Lett. 88, 095003 (2002).
[CrossRef] [PubMed]

H. R. Xia, C. Y. Ye, and S. Y. Zhu, “Experimental observation of spontaneous emission cancellation,” Phys. Rev. Lett. 77, 1032-1034 (1996).
[CrossRef] [PubMed]

M. V. Gurudev, J. Cheng, B. Li, X. D. Xu, X. Q. Li, P. R. Berman, D. G. Steel, A. S. Bracker, and L. J. Sham, “Stimulated and spontaneous optical generation fo electron spin coherence in charged GaAs quantum dots,” Phys. Rev. Lett. 94, 227403 (2005).
[CrossRef]

A. Imamoglu, H. Schmidt, G. Woods, and M. Deutsch, “Strongly interacting photons in a nonlinear cavity,” Phys. Rev. Lett. 79, 1467-1470 (1997).
[CrossRef]

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

S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
[CrossRef]

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, “Measurement of conditions phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710-4714 (1995).
[CrossRef] [PubMed]

H. Kang and Y. F. Zhu, “Observation of large Kerr nonlinearity at low light intensities,” Phys. Rev. Lett. 91, 093601 (2003).
[CrossRef] [PubMed]

Other (3)

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge Univ. Press, 1997).

G. S. Agarwal, Quantum Optics (Springer-Verlag, 1974).

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Progress in Optics, E.Wolf, ed. (Amsterdam, 1996), pp. 257-354.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Schematic diagram of a four-level Y-type atomic system driven by three coherent fields.

Fig. 2
Fig. 2

The self-Kerr nonlinearity Re [ χ ( 3 ) ] (solid curve), linear absorption Im [ χ ( 1 ) ] (dotted curve), and nonlinear absorption Im [ χ ( 3 ) ] (dash-dotted curve) of the four-level Y-type system with different SGC. Parameters are Φ = 2 k × π 2 and Ω c ( 0 ) = 4.0 γ 2 , Ω d ( 0 ) = 1.5 γ 2 , γ 3 = 0.1 γ 2 , γ 4 = 0.01 γ 2 .

Fig. 3
Fig. 3

(a) The self-Kerr nonlinearity Re [ χ ( 3 ) ] relative to probe detuning with different phase Φ. (b) The self-Kerr nonlinearity Re [ χ ( 3 ) ] (solid curve) and nonlinear absorption Im [ χ ( 3 ) ] (dotted curve) relative to probe detuning at Φ = ( 2 k + 1 ) × π 2 . η = 0.997 and other parameters are the same as in Fig. 2.

Fig. 4
Fig. 4

The self-Kerr nonlinearity Re [ χ ( 3 ) ] relative to phase Φ for Δ = 0.13 γ 2 . η = 0.997 and other parameters are the same as in Fig. 2.

Fig. 5
Fig. 5

The extra coherence terms T 1 and T 2 as a function of the probe detuning with different SGC. The main figure represents the effect on the Kerr nonlinearity. The right inset represents the effect on the nonlinear absorption. Φ = 2 k × π 2 and other parameters are the same as in Fig. 2.

Equations (18)

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

H = Δ σ 44 [ Ω d σ 21 + Ω c σ 32 + Ω p σ 42 + H.c. ] + k λ ω k λ b k λ + b k λ [ m = 3 4 k λ g m k σ m 2 b k λ + H.c. ] ,
g m k = ω k λ 2 ϵ 0 V ( μ 2 m e k λ ) ,
d ρ d t = i [ H I , ρ ] + 1 2 l ρ ,
H I = Δ σ 44 [ Ω d σ 21 + Ω c σ 32 + Ω p σ 42 + H.c. ] ,
l = 2 γ 4 [ 2 σ 24 ρ σ 42 σ 44 ρ ρ σ 44 ] + 2 γ 3 [ 2 σ 23 ρ σ 32 σ 33 ρ ρ σ 33 ] + 2 γ 2 [ 2 σ 12 ρ σ 21 σ 22 ρ ρ σ 22 ] + 2 γ 43 [ 2 σ 24 ρ σ 32 σ 34 ρ ρ σ 34 ] + 2 γ 43 [ 2 σ 23 ρ σ 42 σ 43 ρ ρ σ 43 ] .
γ 43 = η γ 4 γ 3 .
ρ ̃ ̇ 12 = γ 2 ρ ̃ 12 + i G d ( ρ ̃ 22 ρ ̃ 11 ) i G c σ 13 i G p ρ ̃ 14 ,
ρ ̃ ̇ 13 = γ 3 ρ ̃ 13 + i G d ρ ̃ 23 i G c ρ ̃ 12 η e i Φ γ 3 γ 4 G 14 ,
ρ ̃ ̇ 14 = ( γ 4 i Δ ) ρ ̃ 14 + i G d ρ ̃ 24 i G p ρ ̃ 12 η e i Φ γ 3 γ 4 ρ ̃ 13 ,
ρ ̃ ̇ 23 = ( γ 2 + γ 3 ) ρ ̃ 23 + i G d ρ ̃ 13 + i G c ( ρ ̃ 33 ρ ̃ 22 ) + i G p ρ ̃ 43 η e i Φ γ 3 γ 4 ρ ̃ 24 ,
ρ ̃ ̇ 24 = ( γ 2 + γ 4 i Δ ) ρ ̃ 24 + i G p ( ρ ̃ 44 ρ ̃ 22 ) + i G d ρ ̃ 14 + i G c ρ ̃ 34 η e i Φ γ 3 γ 4 ρ ̃ 23 ,
ρ ̃ ̇ 34 = ( γ 3 + γ 4 i Δ ) ρ ̃ 34 + i G c ρ ̃ 24 i G p ρ ̃ 32 η e i Φ γ 3 γ 4 ( ρ ̃ 33 + ρ ̃ 44 ) ,
ρ ̃ ̇ 11 = 2 γ 2 ρ ̃ 22 + i G d ( ρ ̃ 21 ρ ̃ 12 ) ,
ρ ̃ ̇ 33 = 2 γ 3 ρ ̃ 33 + i G c ( ρ ̃ 23 ρ ̃ 32 ) η e i Φ γ 3 γ 4 ρ ̃ 34 η e i Φ γ 3 γ 4 ρ ̃ 43 ,
ρ ̃ ̇ 44 = 2 γ 4 ρ ̃ 44 + i G p ( ρ ̃ 24 ρ ̃ 42 ) η e i Φ γ 3 γ 4 ρ ̃ 34 η e i Φ γ 3 γ 4 ρ ̃ 43 .
χ ( 1 ) = 2 N | μ 12 | 2 ϵ 0 G p ρ ̃ 24 ( 1 ) ,
χ ( 3 ) = 2 N | μ 12 | 4 3 ϵ 0 3 G p 3 ρ ̃ 24 ( 3 ) ,
χ = χ ( 1 ) + 3 | E p | 2 χ ( 3 ) .

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