Abstract

The nonlinear behavior of a probe pulse propagating in a medium with electromagnetically induced transparency is studied both numerically and analytically. A new type of nonlinear wave equation is proposed in which the noninstantaneous response of nonlinear polarization is treated properly. The resulting nonlinear behavior of the propagating probe pulse is shown to be fundamentally different from that predicted by the simple nonlinear Schrödinger-like wave equation that considers only instantaneous Kerr nonlinearity.

© 2005 Optical Society of America

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  1. A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
    [CrossRef]
  2. S. E. Harris, Phys. Today 50(7), 36 (1997).
    [CrossRef]
  3. M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001).
    [CrossRef] [PubMed]
  4. L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
    [CrossRef]
  5. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
    [CrossRef] [PubMed]
  6. A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
    [CrossRef]
  7. S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
    [CrossRef] [PubMed]
  8. H. Kane, G. Hernandez, and Y. Zhu, Phys. Rev. A 70, 011801 (2004).
    [CrossRef]
  9. M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
    [CrossRef] [PubMed]
  10. H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. Lett. 87, 073601 (2001).
    [CrossRef]
  11. H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. A 65, 011801 (2002).
    [CrossRef]
  12. T. Hong, Phys. Rev. Lett. 90, 183901 (2003).
    [CrossRef]
  13. X. J. Liu, H. Jing, and M. L. Ge, Phys. Rev. A 70, 055802 (2004).
    [CrossRef]
  14. Y. Wu and L. Deng, Opt. Lett. 29, 2064 (2004).
    [CrossRef] [PubMed]
  15. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).
  16. L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
    [CrossRef]
  17. L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
    [CrossRef]
  18. Y. Wu and L. Deng, Phys. Rev. Lett. 93, 143904 (2004).
    [CrossRef]

2004 (4)

H. Kane, G. Hernandez, and Y. Zhu, Phys. Rev. A 70, 011801 (2004).
[CrossRef]

X. J. Liu, H. Jing, and M. L. Ge, Phys. Rev. A 70, 055802 (2004).
[CrossRef]

Y. Wu and L. Deng, Opt. Lett. 29, 2064 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef]

2003 (2)

S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef] [PubMed]

T. Hong, Phys. Rev. Lett. 90, 183901 (2003).
[CrossRef]

2002 (4)

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. A 65, 011801 (2002).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
[CrossRef]

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

2001 (4)

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
[CrossRef] [PubMed]

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001).
[CrossRef] [PubMed]

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef]

2000 (1)

M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
[CrossRef] [PubMed]

1999 (1)

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

1997 (1)

S. E. Harris, Phys. Today 50(7), 36 (1997).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

Behroozi, C. H.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
[CrossRef] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Bigelow, S.

S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef] [PubMed]

Boyd, R. W.

S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef] [PubMed]

Deng, L.

Y. Wu and L. Deng, Opt. Lett. 29, 2064 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
[CrossRef]

Dutton, Z.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
[CrossRef] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Fleischhauer, M.

M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
[CrossRef] [PubMed]

Ge, M. L.

X. J. Liu, H. Jing, and M. L. Ge, Phys. Rev. A 70, 055802 (2004).
[CrossRef]

Goorskey, D.

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. A 65, 011801 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef]

Hagley, E. W.

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

Harris, S. E.

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

S. E. Harris, Phys. Today 50(7), 36 (1997).
[CrossRef]

Hau, L. V.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
[CrossRef] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

Hernandez, G.

H. Kane, G. Hernandez, and Y. Zhu, Phys. Rev. A 70, 011801 (2004).
[CrossRef]

Hong, T.

T. Hong, Phys. Rev. Lett. 90, 183901 (2003).
[CrossRef]

Imamoglu, A.

M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001).
[CrossRef] [PubMed]

Jing, H.

X. J. Liu, H. Jing, and M. L. Ge, Phys. Rev. A 70, 055802 (2004).
[CrossRef]

Kane, H.

H. Kane, G. Hernandez, and Y. Zhu, Phys. Rev. A 70, 011801 (2004).
[CrossRef]

Kocharovskaya, O.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Kozuma, M.

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
[CrossRef]

Lepeshkin, N. N.

S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef] [PubMed]

Liu, C.

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
[CrossRef] [PubMed]

Liu, X. J.

X. J. Liu, H. Jing, and M. L. Ge, Phys. Rev. A 70, 055802 (2004).
[CrossRef]

Lukin, M. D.

M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001).
[CrossRef] [PubMed]

M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
[CrossRef] [PubMed]

Matsko, A. B.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Payne, M. G.

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

Rostovtsev, Y.

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Rostovtsev, Y. V.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

Scully, M. O.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Wang, H.

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. A 65, 011801 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef]

Welch, G. R.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Wu, Y.

Y. Wu and L. Deng, Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef]

Y. Wu and L. Deng, Opt. Lett. 29, 2064 (2004).
[CrossRef] [PubMed]

Xiao, M.

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. A 65, 011801 (2002).
[CrossRef]

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef]

Zhu, Y.

H. Kane, G. Hernandez, and Y. Zhu, Phys. Rev. A 70, 011801 (2004).
[CrossRef]

Zibrov, A. S.

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Adv. At., Mol., Opt. Phys. (1)

A. B. Matsko, O. Kocharovskaya, Y. Rostovtsev, G. R. Welch, A. S. Zibrov, and M. O. Scully, Adv. At., Mol., Opt. Phys. 46, 191 (2001).
[CrossRef]

Nature (3)

M. D. Lukin and A. Imamoglu, Nature 413, 273 (2001).
[CrossRef] [PubMed]

L. V. Hau, S. E. Harris, Z. Dutton, and C. H. Behroozi, Nature 397, 594 (1999).
[CrossRef]

C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, Nature 409, 490 (2001).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (4)

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. A 65, 011801 (2002).
[CrossRef]

X. J. Liu, H. Jing, and M. L. Ge, Phys. Rev. A 70, 055802 (2004).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. A 65, 051805 (2002).
[CrossRef]

H. Kane, G. Hernandez, and Y. Zhu, Phys. Rev. A 70, 011801 (2004).
[CrossRef]

Phys. Rev. Lett. (6)

M. Fleischhauer and M. D. Lukin, Phys. Rev. Lett. 84, 5094 (2000).
[CrossRef] [PubMed]

H. Wang, D. Goorskey, and M. Xiao, Phys. Rev. Lett. 87, 073601 (2001).
[CrossRef]

A. S. Zibrov, A. B. Matsko, O. Kocharovskaya, Y. V. Rostovtsev, G. R. Welch, and M. O. Scully, Phys. Rev. Lett. 88, 103601 (2002).
[CrossRef]

Y. Wu and L. Deng, Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef]

L. Deng, E. W. Hagley, M. Kozuma, and M. G. Payne, Phys. Rev. Lett. 88, 143902 (2002).
[CrossRef]

T. Hong, Phys. Rev. Lett. 90, 183901 (2003).
[CrossRef]

Phys. Today (1)

S. E. Harris, Phys. Today 50(7), 36 (1997).
[CrossRef]

Science (1)

S. Bigelow, N. N. Lepeshkin, and R. W. Boyd, Science 301, 200 (2003).
[CrossRef] [PubMed]

Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 1995).

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Figures (4)

Fig. 1
Fig. 1

Three-level EIT system in the presence of a cw coupling field and a pulsed probe field with Rabi frequencies of 2 Ω c and 2 Ω p , respectively.

Fig. 2
Fig. 2

Nonlinear polarization z = 0 . Solid curve, calculated from Eq. (9); dotted curve, calculated by direct simulation of Eqs. (1, 2); dashed curve, calculated from the equation in Ref. [14].

Fig. 3
Fig. 3

Propagation of the probe pulse. From left to right, the curves correspond to z = 0 , 24, 48, 72, 96, 120 cm .

Fig. 4
Fig. 4

Another example of the probe pulse propagation, with Ω 0 = 20 × 10 6 s 1 .

Equations (13)

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

Ω p z + 1 c Ω p t = i κ 02 A 2 A 0 * ,
A 0 t = i Ω p * A 2 ,
A 1 t = i ( Δ t + i γ 1 ) A 1 + i Ω c * A 2 ,
A 2 t = i ( Δ s + i γ 2 ) A 2 + i Ω c A 1 + i Ω p A 0 .
F { A 2 } = H ( ω ) F { Ω p A 0 } ,
H ( ω ) = ω + Δ t + i γ 1 Ω c 2 ( ω + Δ t + i γ 1 ) ( ω + Δ s + i γ 2 ) .
A 2 = ( η 0 + i η 1 t η 2 2 t 2 ) ( Ω p , A 0 ) .
A 0 = i t Ω p * ( z , t ) A 2 ( z , t ) d t .
A 2 ( 1 ) = η 0 Ω p + i η 1 Ω p t η 2 2 Ω p t 2 ,
A 0 ( 2 ) = t i Ω p * ( z , t ) A 2 ( 1 ) ( z , t ) d t ,
A 2 ( 3 ) = ( η 0 + i η 1 t η 2 2 t 2 ) [ Ω p A 0 ( 2 ) ] .
i ( Ω p z + 1 V g Ω p t ) + K 0 Ω p + K 2 2 Ω p t 2 = NLT ,
NLT = κ 02 ( A 2 ( 3 ) + A 2 ( 1 ) [ A 0 ( 2 ) ] * ) P ( 3 ) ( z , t ) .

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