Abstract

In nonlinear atomic gaseous media the interference between independent macroscopic polarizations of different ensembles within the atomic velocity distribution can lead to anomalous behavior of the multiwave mixing spectra. For phase-conjugate six-wave mixing (SWM) in a Doppler-broadened N-type four-level system, it is found that the SWM spectrum can be either Doppler-free or very broad, depending on the directions of the incident beams and the ratios between the magnitudes of the wave vectors. This polarization interference can be controlled by the strong coupling field which, on the other hand, can induce Autler–Townes (AT) splitting when the SWM spectrum is Doppler-free. This technique may find special applications as an improved form of AT spectroscopy with higher resolution.

© 2009 Optical Society of America

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  1. S. E. Harris and Y. Yamamoto, “Photon switching by quantum interference,” Phys. Rev. Lett. 81, 3611-3614 (1998).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  4. M. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
    [CrossRef] [PubMed]
  5. 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]
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    [CrossRef] [PubMed]
  7. H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
    [CrossRef]
  8. 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]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  15. Y. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett. 90, 111104 (2007).
    [CrossRef]
  16. Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2007

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]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. 32, 1120-1122 (2007).
[CrossRef] [PubMed]

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Y. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett. 90, 111104 (2007).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

2006

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

2004

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
[CrossRef] [PubMed]

H. Kang, G. Hernandez, and Y. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

2003

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]

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

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[CrossRef]

2001

M. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

2000

M. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

1999

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

1998

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

1996

Anderson, B.

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Andreeva, C.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[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]

Bloch, D.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

Brown, A. W.

Y. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. 32, 1120-1122 (2007).
[CrossRef] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Cartaleva, S.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

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]

Deng, L.

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]

Ducloy, M.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

Fu, G.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

Fu, P.

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

Harris, S. E.

S. E. Harris, “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]

Hernandez, G.

H. Kang, G. Hernandez, and Y. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
[CrossRef] [PubMed]

Imamoglu, A.

M. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

M. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

H. Schmidt and A. Imamoglu, “Giant Kerr nonlinearities obtained by electromagnetically induced transparency,” Opt. Lett. 21, 1936-1938 (1996).
[CrossRef] [PubMed]

Jiang, Q.

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

Kang, H.

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
[CrossRef] [PubMed]

H. Kang, G. Hernandez, and Y. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

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

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[CrossRef]

Khadka, U.

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Liu, X.

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

Lukin, M.

M. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

M. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[CrossRef] [PubMed]

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]

Payne, M. G.

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]

Petrov, L.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

Saltiel, S. M.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

Sarkisyan, D.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

Schmidt, H.

Sun, J.

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

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]

Varzhapetyan, T.

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[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]

Wen, L.

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[CrossRef]

Wu, L. A.

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

Xiao, M.

Y. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett. 90, 111104 (2007).
[CrossRef]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. 32, 1120-1122 (2007).
[CrossRef] [PubMed]

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Yamamoto, Y.

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

Zhang, Y.

Y. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett. 90, 111104 (2007).
[CrossRef]

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Y. Zhang, A. W. Brown, and M. Xiao, “Observation of interference between four-wave mixing and six-wave mixing,” Opt. Lett. 32, 1120-1122 (2007).
[CrossRef] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

Zhu, Y.

H. Kang, G. Hernandez, and Y. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
[CrossRef] [PubMed]

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

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[CrossRef]

Zuo, Z.

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

Appl. Phys. Lett.

Y. Zhang, U. Khadka, B. Anderson, and M. Xiao, “Controlling four-wave and six-wave mixing processes in multilevel atomic systems,” Appl. Phys. Lett. 91, 221108 (2007).
[CrossRef]

Y. Zhang and M. Xiao, “Enhancement of six-wave mixing by atomic coherence in a four-level inverted Y system,” Appl. Phys. Lett. 90, 111104 (2007).
[CrossRef]

Nature

M. Lukin and A. Imamoglu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273-276 (2001).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. A

H. Kang, G. Hernandez, and Y. Zhu, “Superluminal and slow light propagation in cold atoms,” Phys. Rev. A 70, 011801(R) (2004).
[CrossRef]

Z. Zuo, J. Sun, X. Liu, L. A. Wu, and P. Fu, “Two-photon resonant four-wave mixing in a dressed atomic system: polarization interference in a Doppler-broadened system,” Phys. Rev. A 75, 023805 (2007).
[CrossRef]

C. Andreeva, S. Cartaleva, L. Petrov, S. M. Saltiel, D. Sarkisyan, T. Varzhapetyan, D. Bloch, and M. Ducloy, “Saturation effects in the sub-Doppler spectroscopy of cesium vapor confined in an extremely thin cell,” Phys. Rev. A 76, 013837 (2007).
[CrossRef]

H. Kang, L. Wen, and Y. Zhu, “Normal or anomalous dispersion and gain in a resonant coherent medium,” Phys. Rev. A 68, 063806 (2003).
[CrossRef]

Phys. Rev. Lett.

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]

H. Kang, G. Hernandez, and Y. Zhu, “Slow-light six-wave mixing at low light intensities,” Phys. Rev. Lett. 93, 073601 (2004).
[CrossRef] [PubMed]

Y. Zhang, A. W. Brown, and M. Xiao, “Opening four-wave mixing and six-wave mixing channels via dual electromagnetically induced transparency windows,” Phys. Rev. Lett. 99, 123603 (2007).
[CrossRef] [PubMed]

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

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

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

M. Lukin and A. Imamoglu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419-1422 (2000).
[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]

Z. Zuo, J. Sun, X. Liu, Q. Jiang, G. Fu, L. A. Wu, and P. Fu, “Generalized n-photon resonant 2n-wave mixing in an (n+1)-level system with phase-conjugate geometry,” Phys. Rev. Lett. 97, 193904 (2006).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Energy-level diagram for resonant SWM in an N-type four-level system.

Fig. 2
Fig. 2

SWM intensity versus Δ 3 when beam 2 propagates along the forward direction with parameters Δ 1 Γ 30 = Δ 2 Γ 30 = 0 , and ζ 2 = 0.8 for (a) ζ 3 = 0.8 (solid curve), 1.2 (dashed curve), 1.4 (dotted curve), 1.7 (dash-dotted curve) and (b) ζ 3 = 1.8 (solid curve), 1.9 (dashed curve), 2.4 (dotted curve), 3 (dash-dotted curve). The thin solid curve in (a) represents the corresponding result when polarization interference is neglected.

Fig. 3
Fig. 3

SWM intensity versus Δ 1 when beam 2 propagates along the forward direction with parameters Δ 2 Γ 30 = Δ 3 Γ 30 = 0 , and ζ 2 = 0.8 for (a) ζ 3 = 0.8 (solid curve), 1.2 (dashed curve), 1.4 (dotted curve), 1.7 (dash-dotted curve); (b) ζ 3 = 1.8 (solid curve), 1.9 (dashed curve).

Fig. 4
Fig. 4

SWM signal intensity as a function of ζ 3 when beam 2 propagates along the forward direction with Δ 1 Γ 30 = Δ 2 Γ 30 = Δ 3 Γ 30 = 0 , for ζ 2 = 1 (solid curve), 0.8 (dashed curve), 0.6 (dotted curve), and 0.4 (dash-dotted curve).

Fig. 5
Fig. 5

SWM intensity versus Δ 3 when beam 2 propagates along the backward direction with Δ 1 Γ 30 = Δ 2 Γ 30 = 0 for (a) ζ 2 = 0.8 , ζ 3 = 0.7 (solid curve), 1.5 (dashed curve), 2 (dotted curve), and 2.5 (dash-dotted curve); (b) ζ 2 = 0.4 , ζ 3 = 0.2 (solid curve), 0.3 (dashed curve), 0.4 (dotted curve), and 0.5 (dash-dotted curve). The thin solid curve in (b) represents the corresponding result when polarization interference is neglected.

Fig. 6
Fig. 6

Velocity dependence of φ when beam 2 propagates along the forward direction with Δ 1 Γ 30 = Δ 2 Γ 30 = 0 , for (a) Δ 3 Γ 30 = 0 , ζ 2 = 0.8 , ζ 3 = 0.8 (solid curve), 1.8 (dashed curve), 2.4 (dotted curve), 3 (dash-dotted curve); (b) ζ 2 = ζ 3 = 0.8 and Δ 3 Γ 30 = 0 (solid curve), 2 (dashed curve), 5 (dotted curve), and 50 (dash-dotted curve).

Fig. 7
Fig. 7

SWM intensity versus Δ 1 in a homogeneously broadened system with Γ 20 Γ 10 = 0.1 , Γ 30 Γ 10 = 1 , and Δ 2 Γ 30 = Δ 3 Γ 30 = 0 , for G 2 Γ 30 = 0.1 (solid curve), 2 (dashed curve), 5 (dotted curve), and 10 (dash-dotted curve).

Fig. 8
Fig. 8

SWM intensity versus Δ 1 when beam 2 propagates along the backward direction with Δ 2 Γ 30 = Δ 3 Γ 30 = 0 and ζ 2 = ζ 3 = 0.8 , for G 2 Γ 30 = 0.1 (solid curve), 2 (dashed curve), 5 (dash-dotted curve), and 10 (dotted curve).

Fig. 9
Fig. 9

SWM intensity versus Δ 3 when beam 2 propagates along the backward direction with Δ 1 Γ 30 = Δ 2 Γ 30 = 0 and ζ 2 = ζ 3 = 0.8 , for G 2 Γ 30 = 0.1 (solid curve), 2 (dashed curve), 5 (dash-dotted curve), and 10 (dotted curve).

Fig. 10
Fig. 10

(a) SWM intensity versus Δ 3 when beam 2 propagates along the forward direction with ζ 2 = ζ 3 = 0.8 and Δ 1 Γ 30 = Δ 2 Γ 30 = 0 , for G 2 Γ 30 = 0.1 (solid curve), 50 (dashed curve), 100 (dotted curve), and 500 (dash-dotted curve). (b) Corresponding curves of the velocity dependence of φ when Δ 3 Γ 30 = 0 .

Equations (15)

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H = Δ 1 | 1 1 | + ( Δ 1 Δ 2 ) | 2 2 | + ( Δ 1 Δ 2 + Δ 3 ) | 3 3 | [ μ 1 E 1 | 1 0 | + μ 2 E 2 | 2 1 | + μ 3 ( E 3 + E 3 ) | 3 2 | + H.c ] .
ρ 00 ( 0 ) E 1 ρ 10 ( 1 ) E 2 * ρ 20 ( 2 ) E 3 ρ 30 ( 3 ) ( E 3 ) * ρ 20 ( 4 ) E 2 ρ 10 ( 5 ) ,
ρ 10 ( 5 ) = i μ 1 μ 2 2 μ 3 2 5 0 0 0 d τ 1 d τ 2 d τ 5 × E 1 ( t τ 1 τ 2 τ 3 τ 4 τ 5 ) × E 2 * ( t τ 2 τ 3 τ 4 τ 5 ) × E 3 ( t τ 3 τ 4 τ 5 ) × [ E 3 ( t τ 4 τ 5 ) ] * E 2 ( t τ 5 ) × e ( i Δ 1 + Γ 10 ) ( τ 1 + τ 5 ) e [ i ( Δ 1 Δ 2 ) + Γ 20 ] ( τ 2 + τ 4 ) × e [ i ( Δ 1 Δ 2 + Δ 3 ) + Γ 30 ] τ 3 .
ρ 10 ( 5 ) ( r , v ) = i S ( r ) ( i Δ 1 d + Γ 10 ) [ i ( Δ 1 d Δ 2 d ) + Γ 20 ] × 1 [ i ( Δ 1 d Δ 2 d + Δ 3 d ) + Γ 30 ] × 1 [ i ( Δ 1 d Δ 2 d + Δ 3 d Δ 3 d ) + Γ 20 ] × 1 [ i ( Δ 1 d + Δ 3 d Δ 3 d ) + Γ 10 ] .
S ( r ) = G 1 | G 2 | 2 G 3 ( G 3 ) * e i ( k 1 + k 3 k 3 ) r ,
P NL ( r ) = N μ 1 d v W ( v ) ρ 10 ( 5 ) ( r , v ) .
W ( v ) = 1 π u exp [ ( v u ) 2 ] ,
P NL ( r ) = i N μ 1 S ( r ) 0 d τ 1 0 d τ 5 × e ( i Δ 1 + Γ 10 ) ( τ 1 + τ 5 ) e [ i ( Δ 1 Δ 2 ) + Γ 20 ] ( τ 2 + τ 4 ) × e [ i ( Δ 1 Δ 2 + Δ 3 ) + Γ 30 ] τ 3 L ( τ 1 , τ 2 , τ 3 , τ 4 , τ 5 ) ,
L = d v W ( v ) e i k 1 v τ 1 e i ( k 1 k 2 ) v τ 2 × e i ( k 1 k 2 + k 3 ) v τ 3 e i ( k 1 k 2 + k 3 k 3 ) v τ 4 × e i ( k 1 + k 3 k 3 ) v τ 5 .
L = e ( k 1 u 2 ) 2 [ ( τ 1 + τ 5 ) + ( 1 + ζ 2 ) ( τ 2 + τ 4 ) + ( 1 + ζ 2 ζ 3 ) τ 3 ] 2
L = e ( k 1 u 2 ) 2 [ ( τ 1 + τ 5 ) + ( 1 ζ 2 ) ( τ 2 + τ 4 ) + ( 1 ζ 2 ζ 3 ) τ 3 ] 2
( τ 1 + τ 5 ) + ( 1 + ζ 2 ) ( τ 2 + τ 4 ) + ( 1 + ζ 2 ζ 3 ) τ 3 = 0 .
( τ 1 + τ 5 ) + ( 1 ζ 2 ) ( τ 2 + τ 4 ) + ( 1 ζ 2 ζ 3 ) τ 3 = 0 .
d ρ d t = i [ H , ρ ] + ( d ρ d t ) relax .
ρ 10 ( 5 ) ( r , v ) = i S ( r ) { ( i Δ 1 d + Γ 10 ) [ i ( Δ 1 d Δ 2 d ) + Γ 20 ] + | G 2 | 2 } 2 × 1 [ i ( Δ 1 d Δ 2 d + Δ 3 d ) + Γ 30 ] .

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