Abstract

We consider an active-Raman-gain scheme for realizing giant Kerr nonlinearity and superluminal optical solitons in a four-state atomic system with a gain doublet. We show that this scheme, which is fundamentally different from those based on electromagnetically induced transparency (EIT), is capable of working at room temperature and eliminating nearly all attenuation and distortion. We demonstrate that, due to the appearance of a gain spectrum hole induced by the quantum interference effect induced by a signal field, a significant enhancement of Kerr nonlin-earity of probe field can be realized effectively, which can be more than ten times larger than that arrived by the EIT-based scheme with the same energy-level configuration. Based on these important features, we obtain a giant cross-phase modulation effect and hence a stable long-distance propagation of optical solitons, which have superluminal propagating velocity and very low generating power.

© 2010 Optical Society of America

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  1. R. W. Boyd, Nonlinear Optics, 2cd edition (Academic, San Diego, 2003).
  2. J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
    [CrossRef]
  3. Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
    [CrossRef] [PubMed]
  4. C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
    [CrossRef] [PubMed]
  5. C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
    [CrossRef]
  6. S. E. Harris and Y. Yamamoto, "Photon Switching by Quantum Interference," Phys. Rev. Lett. 81, 3611 (1998).
    [CrossRef]
  7. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).
  8. M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633 (2005), and references therein.
    [CrossRef]
  9. Y. Wu and L. Deng, "Ultraslow Optical Solitons in a Cold Four-State Medium," Phys. Rev. Lett. 93, 143904 (2004).
    [CrossRef] [PubMed]
  10. Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064 (2004).
    [CrossRef] [PubMed]
  11. G. Huang, L. Deng, and M. G. Payne, "Dynamics of ultraslow optical solitons in a cold three-state atomic system," Phys. Rev. E 72, 016617 (2005).
    [CrossRef]
  12. Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
    [CrossRef]
  13. G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
    [CrossRef]
  14. L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
    [CrossRef]
  15. C. Hang, G. Huang, and L. Deng, "Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system," Phys. Rev. E 73, 036607 (2006).
    [CrossRef]
  16. C. Hang, G. Huang, and L. Deng, "Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system," Phys. Rev. E 74, 046601 (2006).
    [CrossRef]
  17. Y. Wu and X. Yang, "Giant Kerr nonlinearity and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91, 094104 (2007).
    [CrossRef]
  18. C. Hang and G. Huang, "Weak-light ultraslow vector solitons via electromagnetically induced transparency," Phys. Rev. A 77, 033830 (2008).
    [CrossRef]
  19. W.-X. Yang, J.-M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
    [CrossRef]
  20. C. Hang, V. V. Konotop, and G. Huang, "Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field," Phys. Rev. A 79, 033826 (2009).
  21. R. Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34 (1993).
    [CrossRef]
  22. A. M. Steinberg and R. Y. Chiao, "Dispersionless, highly superluminal propagation in a medium with a gain doublet," Phys. Rev. A 49, 2071 (1994).
    [CrossRef] [PubMed]
  23. L. J. Wang, A. Kuzmich, and P. Pogariu, "Superluminal solitons in a Lambda-type atomic system with two-folded levels," Nature (London) 406, 277 (2000).
    [CrossRef]
  24. A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806 (2001).
    [CrossRef]
  25. A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
    [CrossRef] [PubMed]
  26. A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
    [CrossRef]
  27. M. S. Bigelow, N. N. Lepeshkin, and R.W. Boyd, "Superluminal and slow light propagation in a room temperature solid," Science 301, 200 (2003).
    [CrossRef] [PubMed]
  28. M. D. Stenner, D. J. Gauthier, and M. A. Neifield, "The speed of information in a ‘Fast-light’ optical medium," Nature (London) 425, 695 (2003).
    [CrossRef]
  29. M. D. Stenner and D. J. Gauthier, "Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation," Phys. Rev. A 67, 063801 (2003).
    [CrossRef]
  30. R. G. Ghulghazaryan and Y. P. Malakyan, "Superluminal optical pulse propagation in nonlinear coherent media," Phys. Rev. A 67, 063806 (2003).
    [CrossRef]
  31. K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
    [CrossRef]
  32. L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
    [CrossRef]
  33. E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
    [CrossRef]
  34. G. S. Agarwal and S. Dasgupta, "Superluminal propagation via coherent manipulation of the Raman gain process," Phys. Rev. A 70, 023802 (2004).
    [CrossRef]
  35. A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
    [CrossRef]
  36. M. Janowicz and J. Mostowski, "Superluminal propagation of solitary kink like waves in amplifying media," Phys. Rev. E 73, 046613 (2006).
    [CrossRef]
  37. J. Zhang, G. Hernandez, and Y. Zhu, "Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes," Opt. Lett. 31, 2598 (2006).
    [CrossRef] [PubMed]
  38. K. J. Jiang, L. Deng, and M. G. Payne, "Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium," Phys. Rev. A 76, 033819 (2007).
    [CrossRef]
  39. L. Deng and M. G. Payne, "Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium," Phys. Rev. Lett. 98, 253902 (2007).
    [CrossRef] [PubMed]
  40. E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
    [CrossRef] [PubMed]
  41. Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
    [CrossRef] [PubMed]
  42. L. Thévenaz "Slow and fast light in optical fibres," Nat. Photonics 2, 474-481 (2008).
    [CrossRef]
  43. Y. Wu and X. Yang, "Giant Kerr nonlinearities and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91094104 (2007).
    [CrossRef]
  44. S. Residori, U. Bortolozzo, and J. P. Huignard, "Slow and Fast Light in Liquid Crystal Light Valves," Phys. Rev. Lett. 100, 203603 (2008).
    [CrossRef] [PubMed]
  45. T. Baba, "Slow light in photonic crystals," Nat Photon. 2, 465-473 (2008).
    [CrossRef]
  46. A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
    [CrossRef] [PubMed]
  47. B. D. Clader, Q-Han Park, and J. H. Eberly, "Fast light in fully coherent gain media," Opt. Lett. 31, 2921 (2006).
    [CrossRef] [PubMed]
  48. G. S. Agarwal and T. N. Dey, "Fast light solitons in resonant media," Phys. Rev. A 75, 043806 (2007).
    [CrossRef]
  49. G. Huang, C. Hang, and L. Deng, "Gain-assisted superluminal optical solitons at very low light intensity," Phys. Rev. A 77, 011803 (2008).
    [CrossRef]
  50. H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
    [CrossRef]
  51. The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, see R. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol. 43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, "Controlling the Velocity of Light Pulses," Science 326, 1074 (2009).
  52. H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936 (1996).
    [CrossRef] [PubMed]
  53. A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

2009

C. Hang, V. V. Konotop, and G. Huang, "Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field," Phys. Rev. A 79, 033826 (2009).

2008

C. Hang and G. Huang, "Weak-light ultraslow vector solitons via electromagnetically induced transparency," Phys. Rev. A 77, 033830 (2008).
[CrossRef]

W.-X. Yang, J.-M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
[CrossRef]

S. Residori, U. Bortolozzo, and J. P. Huignard, "Slow and Fast Light in Liquid Crystal Light Valves," Phys. Rev. Lett. 100, 203603 (2008).
[CrossRef] [PubMed]

T. Baba, "Slow light in photonic crystals," Nat Photon. 2, 465-473 (2008).
[CrossRef]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
[CrossRef] [PubMed]

L. Thévenaz "Slow and fast light in optical fibres," Nat. Photonics 2, 474-481 (2008).
[CrossRef]

G. Huang, C. Hang, and L. Deng, "Gain-assisted superluminal optical solitons at very low light intensity," Phys. Rev. A 77, 011803 (2008).
[CrossRef]

H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
[CrossRef]

2007

Y. Wu and X. Yang, "Giant Kerr nonlinearities and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91094104 (2007).
[CrossRef]

G. S. Agarwal and T. N. Dey, "Fast light solitons in resonant media," Phys. Rev. A 75, 043806 (2007).
[CrossRef]

Y. Wu and X. Yang, "Giant Kerr nonlinearity and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91, 094104 (2007).
[CrossRef]

K. J. Jiang, L. Deng, and M. G. Payne, "Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium," Phys. Rev. A 76, 033819 (2007).
[CrossRef]

L. Deng and M. G. Payne, "Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium," Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

2006

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
[CrossRef]

M. Janowicz and J. Mostowski, "Superluminal propagation of solitary kink like waves in amplifying media," Phys. Rev. E 73, 046613 (2006).
[CrossRef]

J. Zhang, G. Hernandez, and Y. Zhu, "Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes," Opt. Lett. 31, 2598 (2006).
[CrossRef] [PubMed]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system," Phys. Rev. E 73, 036607 (2006).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system," Phys. Rev. E 74, 046601 (2006).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
[CrossRef]

B. D. Clader, Q-Han Park, and J. H. Eberly, "Fast light in fully coherent gain media," Opt. Lett. 31, 2921 (2006).
[CrossRef] [PubMed]

2005

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633 (2005), and references therein.
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, "Dynamics of ultraslow optical solitons in a cold three-state atomic system," Phys. Rev. E 72, 016617 (2005).
[CrossRef]

Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
[CrossRef]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
[CrossRef]

2004

Y. Wu and L. Deng, "Ultraslow Optical Solitons in a Cold Four-State Medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064 (2004).
[CrossRef] [PubMed]

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
[CrossRef]

G. S. Agarwal and S. Dasgupta, "Superluminal propagation via coherent manipulation of the Raman gain process," Phys. Rev. A 70, 023802 (2004).
[CrossRef]

2003

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef] [PubMed]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

M. S. Bigelow, N. N. Lepeshkin, and R.W. Boyd, "Superluminal and slow light propagation in a room temperature solid," Science 301, 200 (2003).
[CrossRef] [PubMed]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, "The speed of information in a ‘Fast-light’ optical medium," Nature (London) 425, 695 (2003).
[CrossRef]

M. D. Stenner and D. J. Gauthier, "Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation," Phys. Rev. A 67, 063801 (2003).
[CrossRef]

R. G. Ghulghazaryan and Y. P. Malakyan, "Superluminal optical pulse propagation in nonlinear coherent media," Phys. Rev. A 67, 063806 (2003).
[CrossRef]

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
[CrossRef]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

2001

A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806 (2001).
[CrossRef]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

2000

L. J. Wang, A. Kuzmich, and P. Pogariu, "Superluminal solitons in a Lambda-type atomic system with two-folded levels," Nature (London) 406, 277 (2000).
[CrossRef]

1998

S. E. Harris and Y. Yamamoto, "Photon Switching by Quantum Interference," Phys. Rev. Lett. 81, 3611 (1998).
[CrossRef]

1997

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

1996

H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936 (1996).
[CrossRef] [PubMed]

1995

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

1994

A. M. Steinberg and R. Y. Chiao, "Dispersionless, highly superluminal propagation in a medium with a gain doublet," Phys. Rev. A 49, 2071 (1994).
[CrossRef] [PubMed]

1993

R. Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34 (1993).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal and T. N. Dey, "Fast light solitons in resonant media," Phys. Rev. A 75, 043806 (2007).
[CrossRef]

G. S. Agarwal and S. Dasgupta, "Superluminal propagation via coherent manipulation of the Raman gain process," Phys. Rev. A 70, 023802 (2004).
[CrossRef]

Akulshin, A. M.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
[CrossRef]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

Artoni, M.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Baba, T.

T. Baba, "Slow light in photonic crystals," Nat Photon. 2, 465-473 (2008).
[CrossRef]

Bigelow, M. S.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

M. S. Bigelow, N. N. Lepeshkin, and R.W. Boyd, "Superluminal and slow light propagation in a room temperature solid," Science 301, 200 (2003).
[CrossRef] [PubMed]

Bortolozzo, U.

S. Residori, U. Bortolozzo, and J. P. Huignard, "Slow and Fast Light in Liquid Crystal Light Valves," Phys. Rev. Lett. 100, 203603 (2008).
[CrossRef] [PubMed]

Boyd, R. W.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Boyd, R.W.

M. S. Bigelow, N. N. Lepeshkin, and R.W. Boyd, "Superluminal and slow light propagation in a room temperature solid," Science 301, 200 (2003).
[CrossRef] [PubMed]

Cataliotti, F.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Chiao, R. Y.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

A. M. Steinberg and R. Y. Chiao, "Dispersionless, highly superluminal propagation in a medium with a gain doublet," Phys. Rev. A 49, 2071 (1994).
[CrossRef] [PubMed]

R. Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34 (1993).
[CrossRef]

Cimmino, A.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

Clader, B. D.

B. D. Clader, Q-Han Park, and J. H. Eberly, "Fast light in fully coherent gain media," Opt. Lett. 31, 2921 (2006).
[CrossRef] [PubMed]

Dasgupta, S.

G. S. Agarwal and S. Dasgupta, "Superluminal propagation via coherent manipulation of the Raman gain process," Phys. Rev. A 70, 023802 (2004).
[CrossRef]

Deng, L.

G. Huang, C. Hang, and L. Deng, "Gain-assisted superluminal optical solitons at very low light intensity," Phys. Rev. A 77, 011803 (2008).
[CrossRef]

H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
[CrossRef]

L. Deng and M. G. Payne, "Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium," Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

K. J. Jiang, L. Deng, and M. G. Payne, "Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium," Phys. Rev. A 76, 033819 (2007).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system," Phys. Rev. E 74, 046601 (2006).
[CrossRef]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system," Phys. Rev. E 73, 036607 (2006).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, "Dynamics of ultraslow optical solitons in a cold three-state atomic system," Phys. Rev. E 72, 016617 (2005).
[CrossRef]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
[CrossRef]

Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064 (2004).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow Optical Solitons in a Cold Four-State Medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

Dey, T. N.

G. S. Agarwal and T. N. Dey, "Fast light solitons in resonant media," Phys. Rev. A 75, 043806 (2007).
[CrossRef]

Dogariu, A.

A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806 (2001).
[CrossRef]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

Eberly, J. H.

B. D. Clader, Q-Han Park, and J. H. Eberly, "Fast light in fully coherent gain media," Opt. Lett. 31, 2921 (2006).
[CrossRef] [PubMed]

Ferrari, C.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
[CrossRef] [PubMed]

Fleischhauer, M.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633 (2005), and references therein.
[CrossRef]

Gaeta, A. L.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Gauthier, D. J.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, "The speed of information in a ‘Fast-light’ optical medium," Nature (London) 425, 695 (2003).
[CrossRef]

M. D. Stenner and D. J. Gauthier, "Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation," Phys. Rev. A 67, 063801 (2003).
[CrossRef]

Ghulghazaryan, R. G.

R. G. Ghulghazaryan and Y. P. Malakyan, "Superluminal optical pulse propagation in nonlinear coherent media," Phys. Rev. A 67, 063806 (2003).
[CrossRef]

Grangier, Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Grelu, Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Hagley, E. W.

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
[CrossRef]

Hang, C.

C. Hang, V. V. Konotop, and G. Huang, "Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field," Phys. Rev. A 79, 033826 (2009).

H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
[CrossRef]

G. Huang, C. Hang, and L. Deng, "Gain-assisted superluminal optical solitons at very low light intensity," Phys. Rev. A 77, 011803 (2008).
[CrossRef]

C. Hang and G. Huang, "Weak-light ultraslow vector solitons via electromagnetically induced transparency," Phys. Rev. A 77, 033830 (2008).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system," Phys. Rev. E 73, 036607 (2006).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system," Phys. Rev. E 74, 046601 (2006).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
[CrossRef]

Hannaford, P.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
[CrossRef]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

Harris, S. E.

S. E. Harris and Y. Yamamoto, "Photon Switching by Quantum Interference," Phys. Rev. Lett. 81, 3611 (1998).
[CrossRef]

Hernandez, G.

J. Zhang, G. Hernandez, and Y. Zhu, "Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes," Opt. Lett. 31, 2598 (2006).
[CrossRef] [PubMed]

Hood, C. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

Hou, J.-M.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
[CrossRef]

Huang, G.

C. Hang, V. V. Konotop, and G. Huang, "Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field," Phys. Rev. A 79, 033826 (2009).

C. Hang and G. Huang, "Weak-light ultraslow vector solitons via electromagnetically induced transparency," Phys. Rev. A 77, 033830 (2008).
[CrossRef]

G. Huang, C. Hang, and L. Deng, "Gain-assisted superluminal optical solitons at very low light intensity," Phys. Rev. A 77, 011803 (2008).
[CrossRef]

H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system," Phys. Rev. E 73, 036607 (2006).
[CrossRef]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system," Phys. Rev. E 74, 046601 (2006).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, "Dynamics of ultraslow optical solitons in a cold three-state atomic system," Phys. Rev. E 72, 016617 (2005).
[CrossRef]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
[CrossRef]

Huignard, J. P.

S. Residori, U. Bortolozzo, and J. P. Huignard, "Slow and Fast Light in Liquid Crystal Light Valves," Phys. Rev. Lett. 100, 203603 (2008).
[CrossRef] [PubMed]

Imamoglu, A.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633 (2005), and references therein.
[CrossRef]

H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936 (1996).
[CrossRef] [PubMed]

Janowicz, M.

M. Janowicz and J. Mostowski, "Superluminal propagation of solitary kink like waves in amplifying media," Phys. Rev. E 73, 046613 (2006).
[CrossRef]

Jiang, K.

G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
[CrossRef]

Jiang, K. J.

K. J. Jiang, L. Deng, and M. G. Payne, "Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium," Phys. Rev. A 76, 033819 (2007).
[CrossRef]

Kim, J. B.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

Kim, K.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

Kim, S. K.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

Kimble, H. J.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

Konotop, V. V.

C. Hang, V. V. Konotop, and G. Huang, "Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field," Phys. Rev. A 79, 033826 (2009).

Kuzmich, A.

A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806 (2001).
[CrossRef]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

L. J. Wang, A. Kuzmich, and P. Pogariu, "Superluminal solitons in a Lambda-type atomic system with two-folded levels," Nature (London) 406, 277 (2000).
[CrossRef]

Lange, W.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

Lee, C.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

Lee, R.-K.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
[CrossRef]

Lepeshkin, N. N.

M. S. Bigelow, N. N. Lepeshkin, and R.W. Boyd, "Superluminal and slow light propagation in a room temperature solid," Science 301, 200 (2003).
[CrossRef] [PubMed]

Lezama, A.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
[CrossRef]

Li, H.

H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
[CrossRef]

Li, Y.

C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
[CrossRef]

Lin, Q.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
[CrossRef]

Liu, N.-H.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
[CrossRef]

Ma, L.

C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
[CrossRef]

Mabuchi, H.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

Malakyan, Y. P.

R. G. Ghulghazaryan and Y. P. Malakyan, "Superluminal optical pulse propagation in nonlinear coherent media," Phys. Rev. A 67, 063806 (2003).
[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), and references therein.
[CrossRef]

Martinelli, M.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
[CrossRef] [PubMed]

Melloni, A.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
[CrossRef] [PubMed]

Mikhailov, E. E.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
[CrossRef]

Milonni, P. W.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

Moon, H. S.

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

Morichetti, F.

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
[CrossRef] [PubMed]

Mostowski, J.

M. Janowicz and J. Mostowski, "Superluminal propagation of solitary kink like waves in amplifying media," Phys. Rev. E 73, 046613 (2006).
[CrossRef]

Neifield, M. A.

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, "The speed of information in a ‘Fast-light’ optical medium," Nature (London) 425, 695 (2003).
[CrossRef]

Novikova, I.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
[CrossRef]

Odoulov, S.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef] [PubMed]

Okawachi, Y.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Opat, G. I.

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

Ottaviani, C.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Park, Q-Han

B. D. Clader, Q-Han Park, and J. H. Eberly, "Fast light in fully coherent gain media," Opt. Lett. 31, 2921 (2006).
[CrossRef] [PubMed]

Payne, M. G.

K. J. Jiang, L. Deng, and M. G. Payne, "Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium," Phys. Rev. A 76, 033819 (2007).
[CrossRef]

L. Deng and M. G. Payne, "Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium," Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, "Dynamics of ultraslow optical solitons in a cold three-state atomic system," Phys. Rev. E 72, 016617 (2005).
[CrossRef]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
[CrossRef]

Podivilov, E.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef] [PubMed]

Pogariu, P.

L. J. Wang, A. Kuzmich, and P. Pogariu, "Superluminal solitons in a Lambda-type atomic system with two-folded levels," Nature (London) 406, 277 (2000).
[CrossRef]

Poizat, J.-Ph.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Residori, S.

S. Residori, U. Bortolozzo, and J. P. Huignard, "Slow and Fast Light in Liquid Crystal Light Valves," Phys. Rev. Lett. 100, 203603 (2008).
[CrossRef] [PubMed]

Roch, J.-F.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Sautenkov, V. A.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
[CrossRef]

Schmidt, H.

H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936 (1996).
[CrossRef] [PubMed]

Schweinsberg, A.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Sharping, J. E.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Shumelyuk, A.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef] [PubMed]

Sidorov, A. I.

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
[CrossRef]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

Sinatra, A.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Steinberg, A. M.

A. M. Steinberg and R. Y. Chiao, "Dispersionless, highly superluminal propagation in a medium with a gain doublet," Phys. Rev. A 49, 2071 (1994).
[CrossRef] [PubMed]

Stenner, M. D.

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, "The speed of information in a ‘Fast-light’ optical medium," Nature (London) 425, 695 (2003).
[CrossRef]

M. D. Stenner and D. J. Gauthier, "Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation," Phys. Rev. A 67, 063801 (2003).
[CrossRef]

Sturman, B.

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef] [PubMed]

Thévenaz, L.

L. Thévenaz "Slow and fast light in optical fibres," Nat. Photonics 2, 474-481 (2008).
[CrossRef]

Tombesi, P.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Turchette, Q. A.

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

Vigneron, K.

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Vitali, D.

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

Wang, L. J.

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806 (2001).
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L. J. Wang, A. Kuzmich, and P. Pogariu, "Superluminal solitons in a Lambda-type atomic system with two-folded levels," Nature (London) 406, 277 (2000).
[CrossRef]

Wang, L.-G.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
[CrossRef]

Welch, G. R.

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
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Wu, Y.

Y. Wu and X. Yang, "Giant Kerr nonlinearities and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91094104 (2007).
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Y. Wu and X. Yang, "Giant Kerr nonlinearity and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91, 094104 (2007).
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Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
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Y. Wu and L. Deng, "Ultraslow Optical Solitons in a Cold Four-State Medium," Phys. Rev. Lett. 93, 143904 (2004).
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Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064 (2004).
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Yamamoto, Y.

S. E. Harris and Y. Yamamoto, "Photon Switching by Quantum Interference," Phys. Rev. Lett. 81, 3611 (1998).
[CrossRef]

Yang, W.-X.

W.-X. Yang, J.-M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
[CrossRef]

Yang, X.

Y. Wu and X. Yang, "Giant Kerr nonlinearities and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91094104 (2007).
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Y. Wu and X. Yang, "Giant Kerr nonlinearity and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91, 094104 (2007).
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Zhang, J.

J. Zhang, G. Hernandez, and Y. Zhu, "Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes," Opt. Lett. 31, 2598 (2006).
[CrossRef] [PubMed]

Zhu, S.-Y.

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
[CrossRef]

Zhu, Y.

J. Zhang, G. Hernandez, and Y. Zhu, "Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes," Opt. Lett. 31, 2598 (2006).
[CrossRef] [PubMed]

Zhu, Z.

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

Appl. Phys. Lett.

Y. Wu and X. Yang, "Giant Kerr nonlinearity and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91, 094104 (2007).
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Y. Wu and X. Yang, "Giant Kerr nonlinearities and solitons in a crystal of molecular magnets," Appl. Phys. Lett. 91094104 (2007).
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Nat Photon.

T. Baba, "Slow light in photonic crystals," Nat Photon. 2, 465-473 (2008).
[CrossRef]

Nat. Photonics

L. Thévenaz "Slow and fast light in optical fibres," Nat. Photonics 2, 474-481 (2008).
[CrossRef]

Nature (London)

L. J. Wang, A. Kuzmich, and P. Pogariu, "Superluminal solitons in a Lambda-type atomic system with two-folded levels," Nature (London) 406, 277 (2000).
[CrossRef]

M. D. Stenner, D. J. Gauthier, and M. A. Neifield, "The speed of information in a ‘Fast-light’ optical medium," Nature (London) 425, 695 (2003).
[CrossRef]

Opt. Lett.

J. Zhang, G. Hernandez, and Y. Zhu, "Copropagating superluminal and slow light manifested by electromagnetically assisted nonlinear optical processes," Opt. Lett. 31, 2598 (2006).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow bright and dark optical solitons in a cold three-state medium," Opt. Lett. 29, 2064 (2004).
[CrossRef] [PubMed]

A. Melloni, F. Morichetti, C. Ferrari, and M. Martinelli, "Continuously tunable 1 byte delay in coupled-resonator optical waveguides," Opt. Lett. 33, 2389-2391 (2008).
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B. D. Clader, Q-Han Park, and J. H. Eberly, "Fast light in fully coherent gain media," Opt. Lett. 31, 2921 (2006).
[CrossRef] [PubMed]

H. Schmidt and A. Imamoglu, "Giant Kerr nonlinearities obtained by electromagnetically induced transparency," Opt. Lett. 21, 1936 (1996).
[CrossRef] [PubMed]

Phys. Rev. A

G. S. Agarwal and T. N. Dey, "Fast light solitons in resonant media," Phys. Rev. A 75, 043806 (2007).
[CrossRef]

G. Huang, C. Hang, and L. Deng, "Gain-assisted superluminal optical solitons at very low light intensity," Phys. Rev. A 77, 011803 (2008).
[CrossRef]

H. Li, C. Hang, G. Huang, and L. Deng, "High-order nonlinear Schrödinger equation and superluminal optical solitons in room-temperature active-Raman-gain media," Phys. Rev. A 78, 023822 (2008).
[CrossRef]

C. Hang, Y. Li, L. Ma, and G. Huang, "Three-way entanglement and three-qubit phase gate based on a coherent six-level atomic system," Phys. Rev. A 74, 012319 (2006).
[CrossRef]

C. Hang and G. Huang, "Weak-light ultraslow vector solitons via electromagnetically induced transparency," Phys. Rev. A 77, 033830 (2008).
[CrossRef]

W.-X. Yang, J.-M. Hou, and R.-K. Lee, "Ultraslow bright and dark solitons in semiconductor quantum wells," Phys. Rev. A 77, 033838 (2008).
[CrossRef]

C. Hang, V. V. Konotop, and G. Huang, "Spatial solitons and instabilities of light beams in a three-level atomic medium with a standing-wave control field," Phys. Rev. A 79, 033826 (2009).

R. Y. Chiao, "Superluminal (but causal) propagation of wave packets in transparent media with inverted atomic populations," Phys. Rev. A 48, R34 (1993).
[CrossRef]

A. M. Steinberg and R. Y. Chiao, "Dispersionless, highly superluminal propagation in a medium with a gain doublet," Phys. Rev. A 49, 2071 (1994).
[CrossRef] [PubMed]

Y. Wu, "Two-color ultraslow optical solitons via four-wave mixing in cold-atom media," Phys. Rev. A 71, 053820 (2005).
[CrossRef]

K. J. Jiang, L. Deng, and M. G. Payne, "Superluminal propagation of an optical pulse in a Doppler-broadened three-state single-channel active Raman gain medium," Phys. Rev. A 76, 033819 (2007).
[CrossRef]

E. E. Mikhailov, V. A. Sautenkov, I. Novikova, and G. R. Welch, "Large negative and positive delay of optical pulses in coherently prepared dense Rb vapor with buffer gas," Phys. Rev. A 69, 063808 (2004).
[CrossRef]

G. S. Agarwal and S. Dasgupta, "Superluminal propagation via coherent manipulation of the Raman gain process," Phys. Rev. A 70, 023802 (2004).
[CrossRef]

A. Lezama, A. M. Akulshin, A. I. Sidorov, and P. Hannaford, "Storage and retrieval of light pulses in atomic media with ‘slow’ and ‘fast’ light," Phys. Rev. A 73, 033806 (2006).
[CrossRef]

M. D. Stenner and D. J. Gauthier, "Pump-beam-instability limits to Raman-gain-doublet ‘Fast-light’ pulse propagation," Phys. Rev. A 67, 063801 (2003).
[CrossRef]

R. G. Ghulghazaryan and Y. P. Malakyan, "Superluminal optical pulse propagation in nonlinear coherent media," Phys. Rev. A 67, 063806 (2003).
[CrossRef]

K. Kim, H. S. Moon, C. Lee, S. K. Kim, and J. B. Kim, "Observation of arbitrary group velocities of light from superluminal to subluminal on a single atomic transition line," Phys. Rev. A 68, 013810 (2003).
[CrossRef]

A. Dogariu, A. Kuzmich, and L. J. Wang, "Transparent anomalous dispersion and superluminal light-pulse propagation at a negative group velocity," Phys. Rev. A 63, 053806 (2001).
[CrossRef]

A. M. Akulshin, A. Cimmino, A. I. Sidorov, P. Hannaford, and G. I. Opat, "Light propagation in an atomic medium with steep and sign-reversible dispersion," Phys. Rev. A 67, 011801 (2003).
[CrossRef]

Phys. Rev. E

L.-G. Wang, N.-H. Liu, Q. Lin, and S.-Y. Zhu, "Superluminal propagation of light pulses: A result of interference," Phys. Rev. E 68, 066606 (2003).
[CrossRef]

M. Janowicz and J. Mostowski, "Superluminal propagation of solitary kink like waves in amplifying media," Phys. Rev. E 73, 046613 (2006).
[CrossRef]

G. Huang, K. Jiang, M. G. Payne, and L. Deng, "Formation and propagation of coupled ultraslow optical soliton pairs in a cold three-state double-Lambda system," Phys. Rev. E 73, 056606 (2006).
[CrossRef]

L. Deng, M. G. Payne, G. Huang, and E. W. Hagley, "Formation and propagation of matched and coupled ultraslow optical soliton pairs in a four-level double-lambda system," Phys. Rev. E 72, 055601 (2005).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Generalized nonlinear Schrödinger equation and ultraslow optical solitons in a cold four-state atomic system," Phys. Rev. E 73, 036607 (2006).
[CrossRef]

C. Hang, G. Huang, and L. Deng, "Stable high-dimensional spatial weak-light solitons in a resonant three-state atomic system," Phys. Rev. E 74, 046601 (2006).
[CrossRef]

G. Huang, L. Deng, and M. G. Payne, "Dynamics of ultraslow optical solitons in a cold three-state atomic system," Phys. Rev. E 72, 016617 (2005).
[CrossRef]

Phys. Rev. Lett.

S. E. Harris and Y. Yamamoto, "Photon Switching by Quantum Interference," Phys. Rev. Lett. 81, 3611 (1998).
[CrossRef]

J.-F. Roch, K. Vigneron, Ph. Grelu, A. Sinatra, J.-Ph. Poizat, and Ph. Grangier, "Quantum Nondemolition Measurements using Cold Trapped Atoms," Phys. Rev. Lett. 78, 634 (1997).
[CrossRef]

Q. A. Turchette, C. J. Hood, W. Lange, H. Mabuchi, and H. J. Kimble, "Measurement of Conditional Phase Shifts for Quantum Logic," Phys. Rev. Lett. 75, 4710 (1995).
[CrossRef] [PubMed]

C. Ottaviani, D. Vitali, M. Artoni, F. Cataliotti, and P. Tombesi, "Polarization Qubit Phase Gate in Driven Atomic Media," Phys. Rev. Lett. 90, 197902 (2003).
[CrossRef] [PubMed]

L. Deng and M. G. Payne, "Gain-Assisted Large and Rapidly Responding Kerr Effect using a Room-Temperature Active Raman Gain Medium," Phys. Rev. Lett. 98, 253902 (2007).
[CrossRef] [PubMed]

E. Podivilov, B. Sturman, A. Shumelyuk, and S. Odoulov, "Light Pulse Slowing Down up to 0.025 cm/s by Photorefractive Two-Wave Coupling," Phys. Rev. Lett. 91, 083902 (2003).
[CrossRef] [PubMed]

Y. Okawachi, M. S. Bigelow, J. E. Sharping, Z. Zhu, A. Schweinsberg, D. J. Gauthier, R. W. Boyd, and A. L. Gaeta, "Tunable All-Optical Delays via Brillouin Slow Light in an Optical Fiber," Phys. Rev. Lett. 94, 153902 (2005).
[CrossRef] [PubMed]

A. Kuzmich, A. Dogariu, L. J. Wang, P. W. Milonni, and R. Y. Chiao, "Signal Velocity, Causality, and Quantum Noise in Superluminal Light Pulse Propagation," Phys. Rev. Lett. 86, 3925 (2001).
[CrossRef] [PubMed]

Y. Wu and L. Deng, "Ultraslow Optical Solitons in a Cold Four-State Medium," Phys. Rev. Lett. 93, 143904 (2004).
[CrossRef] [PubMed]

S. Residori, U. Bortolozzo, and J. P. Huignard, "Slow and Fast Light in Liquid Crystal Light Valves," Phys. Rev. Lett. 100, 203603 (2008).
[CrossRef] [PubMed]

Rev. Mod. Phys.

M. Fleischhauer, A. Imamoglu, and J. P. Marangos, "Electromagnetically induced transparency: Optics in coherent media," Rev. Mod. Phys. 77, 633 (2005), and references therein.
[CrossRef]

Science

M. S. Bigelow, N. N. Lepeshkin, and R.W. Boyd, "Superluminal and slow light propagation in a room temperature solid," Science 301, 200 (2003).
[CrossRef] [PubMed]

Other

R. W. Boyd, Nonlinear Optics, 2cd edition (Academic, San Diego, 2003).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic, San Diego, 2003).

The superluminal (or fast light) propagation of an optical pulse means the propagating velocity of the pulse exceeds c or even negative. For a clear illustration of the physical meaning of the superluminal light, see R. W. Boyd and D. J. Gauthier, Progress in Optics (Elsevier Science, Amsterdam, 2002), Vol. 43, Chap. 6, p. 275; R. W. Boyd and D. J. Gauthier, "Controlling the Velocity of Light Pulses," Science 326, 1074 (2009).

A. Jeffery and T. Kawahawa, Asymptotic Method in Nonlinear Wave Theory (Pitman, London, 1982).

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Figures (5)

Fig. 1.
Fig. 1.

(Color online) (a): Energy-level configuration and excitation scheme of the four-state ARG system, which interacts with a strong CW coupling field of angular frequency ω c and Rabi frequency 2Ω C (driving the transition |1⎤ ↔ |3⎤), a weak, pulsed probe field of angular frequency ωp and Rabi frequency 2Ω p (driving the transition |2⎤ ↔ |3⎤), and a CW signal field of angular frequency 2ωs and Rabi frequency Ω s (driving the transition |2⎤ ↔ |4⎤). Δ j (j = 3, 4) are the corresponding detunings. (b): The four-state EIT system with the same energy-level configuration.

Fig. 2.
Fig. 2.

(Color online) (a) The curve of Re[K(ω)] characterizing refractive index of the probe field as a function of ω. (b) The curve of -Im[K(ω)] characterizing the appearance of the gain doublet of the probe field for increasing signal fiels. (c) The curve of Vg /c as a function of Δ4. The solid, dashed, and dot-dashed lines correspond to Ω s = 4.0 × 107 s-1, 2.0 × 107 s-1, and 1.0 × 107 s-1, respectively. The inset shows the detail of the curve of Vg /c with Ω s = 1.0 × 107 s-1. The other parameters are given in the text.

Fig. 3.
Fig. 3.

(Color online) (a): The curve of ϕXPM for the EIT system (solid line) and the curve of -10ϕXPM of the EIT system (dashed line), as functions of Ω s . (b): The curves of ϕXPM (solid line) and ϕ =Re[K 0] (dashed line) versus Ω s for ARG system. The parameters are given in the text.

Fig. 4.
Fig. 4.

(Color online) (a): The wave shape of |Ω p /U 0|2 versus t/τ 0 and z. The initial condition is given by Eq. (18) with Ω s = 2.0 × 107 s-1 and τ 0 = 3.5 × 10-6 s. (b): The wave shape of |Ω p /U 0|2 versus t/τ 0 and z. The initial condition is given by Eq. (18) with Ω s = 1.0 × 107 s-1 and τ 0 = 1.0 × 10-5 s. The other parameters are given in the text. (c): The curves of η(z) = [A(z)/W(z)]/[A(0)/W(0)] - 1 versus the propagation length z with Ω s = 1.0 × 107 s-1 and 2.0 × 107 s-1.

Fig. 5.
Fig. 5.

(Color online) The wave shape of |Ω p /U 0|2 versus t/τ 0 and z with the initial condition given by Ω p /U 0 = 0.8sech(σ - 1.0)exp( 1) + 1.2sech(σ + 3.0)exp( 2). Panel (a) and (b) show the collision for Δθθ 2 - θ 1 =0 and π/2. The lower panel show the corresponding contour maps. Both solitons in each panel have resumed their original shapes after their collision.

Tables (2)

Tables Icon

Table 1. Expressions of linear and nonlinear optical susceptibilities and XPM phase shift for the four-state ARG system (Fig. 1(a)) and the four-state EIT system (Fig. 1(b)).

Tables Icon

Table 2. Values of linear and nonlinear optical susceptibilities and XPM phase shift for the four-state ARG scheme and the four-state EIT scheme. The parameters used have been given in the text

Equations (40)

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

H ̂ int / h ¯ = [ Δ 3 3 3 + Δ 4 4 4 + Ω c 3 1 + Ω p 3 2
+ Ω s 4 2 + H . c . ] ,
( i t + d 2 ) A 2 + Ω p * A 3 + Ω s * A 4 = 0 ,
( i t + d 3 ) A 3 + Ω c A 1 + Ω p A 2 = 0 ,
( i t + d 4 ) A 4 + Ω s A 2 = 0
i ( z + 1 c t ) Ω p + κ A 3 A 2 * = 0 ,
K ( ω ) = ω c + κ ( ω d 4 * ) Ω c 2 [ ( ω d 2 * ) ( ω d 4 * ) Ω s 2 ] ( d 3 2 + Ω 2 2 ) .
Ω p ( z , t ) = Ω p ( 0,0 ) e i K 0 z b 1 ( z ) i b 2 ( z ) exp [ ( K 1 t z ) 2 2 [ b 1 ( z ) i b 2 ( z ) ] τ 0 2 ] ,
K 0 = ϕ + i α 2 , K 1 = 1 c κ Ω c 2 Ω s 2 d 3 2 + κ Ω c 2 d 4 * Ω s 4 d 3 2 ,
K 2 = 4 κ Ω c 2 d 4 * Ω s 4 d 3 2 + 2 κ Ω c 2 d 4 * Ω s 6 d 3 2 ,
D ARG = Im K ( ω = ω max ) Im K ( ω = 0 )
= 4 κ Ω s 2 Ω c 2 γ 4 ( 4 Ω s 2 γ 4 2 ) d 3 2 κ γ 4 Ω c 2 Ω s 2 d 3 2 .
Ω s 2 2 κ γ 4 Ω c 2 / d 3 2 .
Γ ARG = 2 4 Ω s 2 2 γ 4 2 + 2 γ 4 4 Ω s 2 γ 4 2 .
Ω p * A 3 ( 0 ) + Ω s * A 4 = ω i γ 2 κ A 3 * ( 0 ) ( K * ω c ) F * exp ( i θ * ) ,
χ p = N a p 23 2 ε 0 h ¯ A 3 A 2 * Ω p χ p ( 1 ) + χ pp ( 3 ) p 2 + χ ps ( 3 ) s 2 ,
A 2 = d 4 Ω c Ω p * D A 1 ,
A 3 = Ω c Ω s D A 1 ,
A 4 = Ω c Ω s Ω p * D A 1 ,
A 1 = [ 1 + Ω c 2 D s 2 + ( Ω s 2 + d 4 2 ) Ω p 2 D 2 ] 1 / 2 .
χ p ( 1 ) = N a p 23 2 ε 0 h ̄ i Ω c 2 γ 2 ( Ω c 2 + d 3 2 ) ,
χ pp ( 3 ) = N a p 23 4 ε 0 h ¯ 3 i ( 2 γ 2 γ 3 + Ω c 2 ) Ω c 2 γ 2 3 ( Ω c 2 + d 3 2 ) 2 ,
χ ps ( 3 ) = N a p 23 2 p 24 2 ε 0 h ¯ 3 d 4 Ω c 2 γ 2 2 ( Ω c 2 + d 3 2 ) d 4 2 ,
ϕ XPM = ω p L 2 c Re [ χ ps ( 3 ) ] s 2 κL Δ 4 Ω c 2 γ 2 2 d 3 2 d 4 2 Ω s 2 ,
i F z 2 K 2 2 2 F τ 2 W e α ̄ z 2 F F 2 = 0 ,
W = κ ( ω d 4 * ) ( ω d 2 * ) ( ω d 4 * ) Ω s 2 ( A 3 * ( 0 ) B 3 ( 2 ) + A 3 ( 0 ) B 3 * ( 2 ) ) ,
i u s + 2 u σ 2 + 2 u u 2 = i L D L A u ,
Ω p = 1 τ 0 K ˜ 2 W ˜ sech [ 1 τ 0 ( t z V g ) ] e i K ˜ 0 z + i z 2 L D ,
V g = 5.4 × 10 5 c .
Ω p = 0.8 U 0 sech ( σ 1.0 ) e i θ 1 + 1.2 U 0 sech ( σ + 3.0 ) e i θ 2 ,
A 1 ( 1 ) = A 3 ( 1 ) = 0 ,
A 2 ( 1 ) = [ ( K * ω / c ) F * κ A 3 * ( 0 ) ] e i θ * ,
A 4 ( 1 ) = B 4 ( 1 ) F * e i θ *
A j ( 2 ) = B j ( 2 ) F 2 e α ̄ z 2 , ( j = 1,3 ) ,
A 2 ( 2 ) = i κ A 3 * ( 0 ) ( 1 c 1 V g * ) F * t 1 e i θ * ,
A 4 ( 2 ) = i κ A 3 * ( 0 ) Ω s * [ K * ω c + ( ω + d 2 ) ( 1 V g * 1 c ) ]
× F * t 1 e i θ * ,
B 1 ( 2 ) = 1 2 A 1 ( 0 ) ( 1 + Ω c / d 3 2 ) [ K ω / c κ d 3 * + K * ω / c κ d 3
K ω / c 2 κ 2 A 3 ( 0 ) 2 B 4 ( 1 ) 2 ] ,
B 3 ( 2 ) = 1 d 3 ( K * ω / c κ A 3 * ( 0 ) + Ω c B 1 ( 2 ) ) .

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