Abstract

Four-level polarization beats (FLPBs) with broadband noisy light are investigated by use of chaotic field, phase-diffusion, and Gaussian-amplitude models. The polarization beat signal is shown to be particularly sensitive to the statistical properties of the Markovian stochastic light fields with arbitrary bandwidth. Different stochastic models of the laser field affect only the fourth-order coherence functions. The constant background of the beat signal originates from the amplitude fluctuation of the Markovian stochastic fields. The Gaussian-amplitude field shows fluctuations larger than the chaotic field, which in turn exhibits fluctuations much larger than for the phase-diffusion field with pure phase fluctuations caused by spontaneous emission. It is also found that the asymmetric behaviors of the polarization beat signals due to the unbalanced dispersion effects between the two arms of an interferometer and to the Doppler width do not affect the overall accuracy when FLPBs are used to measure the energy-level difference between two states that are dipolar forbidden from the ground state.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. E. Ryan and T. H. Bergeman, “Hanle effect in nonmonochromatic laser light,” Phys. Rev. A 43, 6142–6155 (1991).
    [CrossRef] [PubMed]
  2. C. Chen, D. S. Elliott, and M. W. Hamilton, “Two-photon absorption from the real Gaussian field,” Phys. Rev. Lett. 68, 3531–3534 (1992).
    [CrossRef] [PubMed]
  3. R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
    [CrossRef] [PubMed]
  4. M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
    [CrossRef] [PubMed]
  5. R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
    [CrossRef] [PubMed]
  6. A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034–2049 (1980).
    [CrossRef]
  7. R. Bratfalean and P. Ewart, “Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields,” Phys. Rev. A 56, 2267–2279 (1997).
    [CrossRef]
  8. H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarizations in semiconductor doped glasses,” Phys. Rev. Lett. 71, 3649–3652 (1993).
    [CrossRef] [PubMed]
  9. H. Ma, L. H. Acioli, A. S. L. Gomes, and C. B. de Araujo, “Method to determine the phase dispersion of the third-order susceptibility,” Opt. Lett. 16, 630–632 (1991).
    [CrossRef] [PubMed]
  10. Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
    [CrossRef]
  11. Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).
  12. Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Fourth-order interference on polarization beats in a four-level system,” J. Opt. Soc. Am. B 17, 690–696 (2000).
    [CrossRef]
  13. Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
    [CrossRef]
  14. Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, “Higher-order correlation on polarization beats in Markovian stochastic fields,” Phys. Rev. A 63, 043802 (2001).
    [CrossRef]
  15. Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
    [CrossRef]
  16. Y. P. Zhang, T. T. Tang, and P. M. Fu, “Asymmetric behavior of the polarization beats signal in a four-level system,” Acta Phys. Sin. 48, 242–248 (1999).
  17. D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
    [CrossRef] [PubMed]
  18. P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
    [CrossRef] [PubMed]
  19. N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
    [CrossRef]
  20. S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
    [CrossRef]
  21. D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
    [CrossRef]
  22. V. Kozich, L. D. Menezes, and C. B. de Araujo, “Interference effects in time-delayed degenerate four-wave mixing with broadband noisy light,” J. Opt. Soc. Am. B 17, 973–976 (2000).
    [CrossRef]
  23. J. C. Kirkwood, A. C. Albrecht, D. J. Ulness, “Fifth-order nonlinear Raman processes in molecular liquids using quasi-cw noisy light. I. Theory,” J. Chem. Phys. 111, 253–271 (1999).
    [CrossRef]
  24. J. C. Kirkwood and A. C. Albrecht, “Down-conversion of electronic frequencies and their dephasing dynamics: Interferometric four-wave-mixing spectroscopy with broadband light,” Phys. Rev. A 61, 033802 (2000).
    [CrossRef]
  25. R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am. B 3, 1295–1304 (1986).
    [CrossRef]
  26. Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
    [CrossRef]
  27. Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
    [CrossRef]
  28. Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
    [CrossRef]
  29. Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
    [CrossRef]
  30. Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
    [CrossRef]
  31. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1981).

2001 (2)

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, “Higher-order correlation on polarization beats in Markovian stochastic fields,” Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

2000 (10)

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Fourth-order interference on polarization beats in a four-level system,” J. Opt. Soc. Am. B 17, 690–696 (2000).
[CrossRef]

V. Kozich, L. D. Menezes, and C. B. de Araujo, “Interference effects in time-delayed degenerate four-wave mixing with broadband noisy light,” J. Opt. Soc. Am. B 17, 973–976 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
[CrossRef]

J. C. Kirkwood and A. C. Albrecht, “Down-conversion of electronic frequencies and their dephasing dynamics: Interferometric four-wave-mixing spectroscopy with broadband light,” Phys. Rev. A 61, 033802 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
[CrossRef]

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
[CrossRef]

1999 (3)

Y. P. Zhang, T. T. Tang, and P. M. Fu, “Asymmetric behavior of the polarization beats signal in a four-level system,” Acta Phys. Sin. 48, 242–248 (1999).

Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).

J. C. Kirkwood, A. C. Albrecht, D. J. Ulness, “Fifth-order nonlinear Raman processes in molecular liquids using quasi-cw noisy light. I. Theory,” J. Chem. Phys. 111, 253–271 (1999).
[CrossRef]

1997 (2)

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

R. Bratfalean and P. Ewart, “Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields,” Phys. Rev. A 56, 2267–2279 (1997).
[CrossRef]

1995 (2)

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
[CrossRef] [PubMed]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

1993 (2)

H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarizations in semiconductor doped glasses,” Phys. Rev. Lett. 71, 3649–3652 (1993).
[CrossRef] [PubMed]

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

1992 (2)

C. Chen, D. S. Elliott, and M. W. Hamilton, “Two-photon absorption from the real Gaussian field,” Phys. Rev. Lett. 68, 3531–3534 (1992).
[CrossRef] [PubMed]

R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
[CrossRef] [PubMed]

1991 (2)

1986 (2)

1984 (2)

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

1980 (1)

A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034–2049 (1980).
[CrossRef]

Acioli, L. H.

Albrecht, A. C.

J. C. Kirkwood and A. C. Albrecht, “Down-conversion of electronic frequencies and their dephasing dynamics: Interferometric four-wave-mixing spectroscopy with broadband light,” Phys. Rev. A 61, 033802 (2000).
[CrossRef]

J. C. Kirkwood, A. C. Albrecht, D. J. Ulness, “Fifth-order nonlinear Raman processes in molecular liquids using quasi-cw noisy light. I. Theory,” J. Chem. Phys. 111, 253–271 (1999).
[CrossRef]

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

Anderson, M. H.

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

Asaka, S.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Beach, R.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Bergeman, T. H.

R. E. Ryan and T. H. Bergeman, “Hanle effect in nonmonochromatic laser light,” Phys. Rev. A 43, 6142–6155 (1991).
[CrossRef] [PubMed]

Blumel, R.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
[CrossRef] [PubMed]

Bratfalean, R.

R. Bratfalean and P. Ewart, “Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields,” Phys. Rev. A 56, 2267–2279 (1997).
[CrossRef]

Chen, C.

C. Chen, D. S. Elliott, and M. W. Hamilton, “Two-photon absorption from the real Gaussian field,” Phys. Rev. Lett. 68, 3531–3534 (1992).
[CrossRef] [PubMed]

Cooper, J.

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
[CrossRef] [PubMed]

de Araujo, C. B.

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, “Higher-order correlation on polarization beats in Markovian stochastic fields,” Phys. Rev. A 63, 043802 (2001).
[CrossRef]

V. Kozich, L. D. Menezes, and C. B. de Araujo, “Interference effects in time-delayed degenerate four-wave mixing with broadband noisy light,” J. Opt. Soc. Am. B 17, 973–976 (2000).
[CrossRef]

H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarizations in semiconductor doped glasses,” Phys. Rev. Lett. 71, 3649–3652 (1993).
[CrossRef] [PubMed]

H. Ma, L. H. Acioli, A. S. L. Gomes, and C. B. de Araujo, “Method to determine the phase dispersion of the third-order susceptibility,” Opt. Lett. 16, 630–632 (1991).
[CrossRef] [PubMed]

DeBeer, D.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

DeMott, D. C.

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

Elliott, D. S.

C. Chen, D. S. Elliott, and M. W. Hamilton, “Two-photon absorption from the real Gaussian field,” Phys. Rev. Lett. 68, 3531–3534 (1992).
[CrossRef] [PubMed]

Ewart, P.

R. Bratfalean and P. Ewart, “Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields,” Phys. Rev. A 56, 2267–2279 (1997).
[CrossRef]

Eyler, E. E.

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, “Higher-order correlation on polarization beats in Markovian stochastic fields,” Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Fu, P. M.

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Fourth-order interference on polarization beats in a four-level system,” J. Opt. Soc. Am. B 17, 690–696 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, and P. M. Fu, “Asymmetric behavior of the polarization beats signal in a four-level system,” Acta Phys. Sin. 48, 242–248 (1999).

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Fujiwara, M.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Georges, A. T.

A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034–2049 (1980).
[CrossRef]

Gomes, A. S. L.

Gustafson, E. K.

Hamilton, M. W.

C. Chen, D. S. Elliott, and M. W. Hamilton, “Two-photon absorption from the real Gaussian field,” Phys. Rev. Lett. 68, 3531–3534 (1992).
[CrossRef] [PubMed]

Hartmann, S. R.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Hou, X.

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
[CrossRef]

Jiang, Q.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Kirkwood, J. C.

J. C. Kirkwood and A. C. Albrecht, “Down-conversion of electronic frequencies and their dephasing dynamics: Interferometric four-wave-mixing spectroscopy with broadband light,” Phys. Rev. A 61, 033802 (2000).
[CrossRef]

J. C. Kirkwood, A. C. Albrecht, D. J. Ulness, “Fifth-order nonlinear Raman processes in molecular liquids using quasi-cw noisy light. I. Theory,” J. Chem. Phys. 111, 253–271 (1999).
[CrossRef]

Kozich, V.

Li, C. S.

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Li, S.

Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).

Li, X. F.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Lu, K. Q.

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
[CrossRef]

Ma, H.

H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarizations in semiconductor doped glasses,” Phys. Rev. Lett. 71, 3649–3652 (1993).
[CrossRef] [PubMed]

H. Ma, L. H. Acioli, A. S. L. Gomes, and C. B. de Araujo, “Method to determine the phase dispersion of the third-order susceptibility,” Opt. Lett. 16, 630–632 (1991).
[CrossRef] [PubMed]

Matsuoke, M.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Menezes, L. D.

Metcalf, H. J.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
[CrossRef] [PubMed]

Mi, X.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Morita, N.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

Nakatsuka, M.

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Ritsch, H.

R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
[CrossRef] [PubMed]

Ryan, R. E.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
[CrossRef] [PubMed]

R. E. Ryan and T. H. Bergeman, “Hanle effect in nonmonochromatic laser light,” Phys. Rev. A 43, 6142–6155 (1991).
[CrossRef] [PubMed]

Siegman, A. E.

Smith, S. J.

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

Sun, L. Q.

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Fourth-order interference on polarization beats in a four-level system,” J. Opt. Soc. Am. B 17, 690–696 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).

Tang, T. T.

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Fourth-order interference on polarization beats in a four-level system,” J. Opt. Soc. Am. B 17, 690–696 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, and P. M. Fu, “Asymmetric behavior of the polarization beats signal in a four-level system,” Acta Phys. Sin. 48, 242–248 (1999).

Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).

Trebino, R.

Ulness, D. J.

J. C. Kirkwood, A. C. Albrecht, D. J. Ulness, “Fifth-order nonlinear Raman processes in molecular liquids using quasi-cw noisy light. I. Theory,” J. Chem. Phys. 111, 253–271 (1999).
[CrossRef]

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

Van Wagenen, L. G.

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Vemuri, G.

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

Walser, R.

R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
[CrossRef] [PubMed]

Wang, P. F.

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Westling, L. A.

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
[CrossRef] [PubMed]

Wu, H. C.

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
[CrossRef]

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
[CrossRef]

Xu, J.

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Yajima, T.

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

Yu, Z. H.

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Zhang, L.

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

Zhang, Y. P.

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, “Higher-order correlation on polarization beats in Markovian stochastic fields,” Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Fourth-order interference on polarization beats in a four-level system,” J. Opt. Soc. Am. B 17, 690–696 (2000).
[CrossRef]

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Y. P. Zhang, T. T. Tang, and P. M. Fu, “Asymmetric behavior of the polarization beats signal in a four-level system,” Acta Phys. Sin. 48, 242–248 (1999).

Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

Zoller, P.

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
[CrossRef] [PubMed]

Acta Phys. Sin. (2)

Y. P. Zhang, T. T. Tang, and P. M. Fu, “Asymmetric behavior of the polarization beats signal in a four-level system,” Acta Phys. Sin. 48, 242–248 (1999).

Y. P. Zhang, T. T. Tang, S. Li, and L. Q. Sun, “Theoretical study of the fifth-order polarization beats in a four-level system,” Acta Phys. Sin. 48, 1452–1458 (1999).

Chin. Phys. (3)

Y. P. Zhang, H. C. Wu, P. F. Wang, C. S. Li, and P. M. Fu, “Phase fluctuations on Raman-enhanced nondegenerate four-wave mixing,” Chin. Phys. 9, 599–605 (2000).
[CrossRef]

Y. P. Zhang, K. Q. Lu, H. C. Wu, J. Xu, and P. M. Fu, “Nonlinear effects of third- and fifth-order in polarization beats,” Chin. Phys. 9, 606–610 (2000).
[CrossRef]

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Raman-enhanced polarization beats with phase-conjugation geometry,” Chin. Phys. 9, 905–909 (2000).
[CrossRef]

Chin. Phys. Lett. (1)

Y. P. Zhang, L. Q. Sun, T. T. Tang, L. Zhang, and P. M. Fu, “Fourth-order interference on polarization beats with phase-conjugation geometry,” Chin. Phys. Lett. 17, 206–208 (2000).
[CrossRef]

J. Chem. Phys. (1)

J. C. Kirkwood, A. C. Albrecht, D. J. Ulness, “Fifth-order nonlinear Raman processes in molecular liquids using quasi-cw noisy light. I. Theory,” J. Chem. Phys. 111, 253–271 (1999).
[CrossRef]

J. Mod. Opt. (1)

Y. P. Zhang, K. Q. Lu, C. S. Li, X. Hou, and C. B. de Araujo, “Correlation effects of chaotic and phase-diffusion fields on polarization beats in a V-type three-level system,” J. Mod. Opt. 48, 549–564 (2001).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

Y. P. Zhang, X. Hou, K. Q. Lu, and H. C. Wu, “Sixth-order correlation on Raman-enhanced polarization beats with phase-conjugation geometry,” Opt. Commun. 184, 265–276 (2000).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (14)

J. C. Kirkwood and A. C. Albrecht, “Down-conversion of electronic frequencies and their dephasing dynamics: Interferometric four-wave-mixing spectroscopy with broadband light,” Phys. Rev. A 61, 033802 (2000).
[CrossRef]

P. M. Fu, X. Mi, Z. H. Yu, Q. Jiang, Y. P. Zhang, and X. F. Li, “Ultrafast modulation spectroscopy in a cascade three-level system,” Phys. Rev. A 52, 4867–4870 (1995).
[CrossRef] [PubMed]

N. Morita and T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A 30, 2525–2536 (1984).
[CrossRef]

S. Asaka, M. Nakatsuka, M. Fujiwara, and M. Matsuoke, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

D. C. DeMott, D. J. Ulness, and A. C. Albrecht, “Femtosecond temporal probes using spectrally tailored noisy quasi-cw laser light,” Phys. Rev. A 55, 761–771 (1997).
[CrossRef]

Y. P. Zhang, L. Q. Sun, T. T. Tang, and P. M. Fu, “Effects of field correlation on polarization beats,” Phys. Rev. A 61, 053819 (2000).
[CrossRef]

R. Walser, H. Ritsch, P. Zoller, and J. Cooper, “Laser-noise-induced population fluctuations in two-level systems: Complex and real Gaussian driving fields,” Phys. Rev. A 45, 468–476 (1992).
[CrossRef] [PubMed]

M. H. Anderson, G. Vemuri, J. Cooper, P. Zoller, and S. J. Smith, “Experimental study of absorption and gain by two-level atoms in a time-delayed non-Markovian optical field,” Phys. Rev. A 47, 3202–3209 (1993).
[CrossRef] [PubMed]

R. E. Ryan, L. A. Westling, R. Blumel, and H. J. Metcalf, “Two-photon spectroscopy: A technique for characterizing diode-laser noise,” Phys. Rev. A 52, 3157–3169 (1995).
[CrossRef] [PubMed]

A. T. Georges, “Resonance fluorescence in Markovian stochastic fields,” Phys. Rev. A 21, 2034–2049 (1980).
[CrossRef]

R. Bratfalean and P. Ewart, “Spectral line shape of nonresonant four-wave mixing in Markovian stochastic fields,” Phys. Rev. A 56, 2267–2279 (1997).
[CrossRef]

R. E. Ryan and T. H. Bergeman, “Hanle effect in nonmonochromatic laser light,” Phys. Rev. A 43, 6142–6155 (1991).
[CrossRef] [PubMed]

Y. P. Zhang, T. T. Tang, L. Q. Sun, and P. M. Fu, “Effects of fourth-order coherence on ultrafast modulation spectroscopy,” Phys. Rev. A 61, 023809 (2000).
[CrossRef]

Y. P. Zhang, C. B. de Araujo, and E. E. Eyler, “Higher-order correlation on polarization beats in Markovian stochastic fields,” Phys. Rev. A 63, 043802 (2001).
[CrossRef]

Phys. Rev. Lett. (3)

C. Chen, D. S. Elliott, and M. W. Hamilton, “Two-photon absorption from the real Gaussian field,” Phys. Rev. Lett. 68, 3531–3534 (1992).
[CrossRef] [PubMed]

H. Ma and C. B. de Araujo, “Interference between third- and fifth-order polarizations in semiconductor doped glasses,” Phys. Rev. Lett. 71, 3649–3652 (1993).
[CrossRef] [PubMed]

D. DeBeer, L. G. Van Wagenen, R. Beach, and S. R. Hartmann, “Ultrafast modulation spectroscopy,” Phys. Rev. Lett. 56, 1128–1131 (1986).
[CrossRef] [PubMed]

Other (1)

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North Holland, Amsterdam, 1981).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Four-level configuration to be treated by FLPB.

Fig. 2
Fig. 2

Schematic diagram of the geometry of FLPB.

Fig. 3
Fig. 3

Beat signal intensity versus relative time delay. The parameters are Ω3-Ω2=254 ps-1, Ω3=3317 ps-1, Δk=0, η=1, Bi=0.6, Γ20=12.5 ps-1, Γ30=14.5 ps-1. δτ=0 fs for dotted curve, δτ=43 fs for dashed curve and δτ=100 fs for solid curve.

Fig. 4
Fig. 4

Three-dimensional interferogram of the beat signal intensity I(τ, r) versus time delay τ and transverse distance r for the Gaussian-amplitude field. The parameters are Ω3-Ω2=254 ps-1, Δk=22.22 mm-1, Γ20a-Γ10=12.5 ps-1, Γ30a-Γ10=14.5 ps-1, η=1, ξi=1.5, Δ1=0, Ai=0.6. δτ=0 fs for (a) and δτ=43 fs for (b).

Fig. 5
Fig. 5

Beat signal intensity versus relative time delay. Solid curves, chaotic field; dashed curve, phase-diffusion field; dotted curve, Gaussian-amplitude field. The parameters are Ω3-Ω2=254 ps-1, Δk=0, η=1, ξi=1.5, Δ1=0, Ai=0.6, δτ=43 fs, Γ20a-Γ10=2.5 ps-1 and Γ30a-Γ10=2.9 ps-1.

Fig. 6
Fig. 6

Spectrum of FWM when beams 2 and 3 consist of ω2 only in which center wavelength is 615.4 nm. Solid curve, theoretical curve with α2=2.5 ps-1 and ω2=3063 ps-1.

Fig. 7
Fig. 7

FWM signal intensity versus relative time delay when beams 2 and 3 consist of ω2 only. Filled squares, experimental data; solid curve, fourth-order theory with α2=2.5 ps-1 and Bi=0.6; dashed curve, second-order theory.

Fig. 8
Fig. 8

Spectrum of FWM when beams 2 and 3 consist of ω3 only in which center wavelength is 568.2 nm. Solid curve, theoretical curve with α3=2.9 ps-1 and ω3=3317 ps-1.

Fig. 9
Fig. 9

FWM signal intensity versus relative time delay when beams 2 and 3 consist of only ω3. Filled squares experimental data; solid curve, fourth-order theory with α3=2.9 ps-1, δτ=43 fs, Bi=0.6 and η=1; dashed curve, second-order theory.

Fig. 10
Fig. 10

Beat signal intensity versus relative time delay. Filled squares, experimental result; solid curve, theoretical curve given by relation (10) with α2=2.5 ps-1, α3=2.9 ps-1, ω3-ω2=2.54×1014 s-1, ω3=3317 ps-1, Δk=0, δτ=43 fs, Bi=0.6, and η=1.

Fig. 11
Fig. 11

Fourier spectrum of the experimental data in which τ is varied over a range of 362 fs.

Equations (64)

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

Ep2(r, t)=A2(r, t)exp(-iω2t)+A3(r, t)exp(-iω3t)=2u2(t)exp[i(k2·r-ω2t)]+3u3(t)exp[i(k3·r-ω3t)],
EP3(r, t)=A2(r, t)exp(-iω2t)+A3(r, t)exp(-iω3t),=2u2(t-τ)exp[i(k2·r-ω2t+ω2τ)]+3u3(t-τ+δτ)exp[i(k3·r-ω3t+ω3τ-ω3δτ)].
EP1(r, t)=A1(r, t)exp(-iω1t)=1u1(t)exp[i(k1·r-ω1t)],
(I)ρ00(0)ω1ρ10(1)ω2ρ20(2)-ω2ρ10(3),
(II)ρ00(0)ω1ρ10(1)ω3ρ30(2)-ω3ρ10(3),
ρ(I)=-iμ1μ223exp(-iω1t)×-+dνw(ν)0dt30dt20dt1×H1(t1)H2(t2)H1(t3)A1(t-t1-t2-t3)×A2(t-t2-t3)[A2(t-t3-τ)]*,
ρ(II)=-iμ1μ323exp(-iω1t)×-+dν w(ν)0dt30dt20dt1×H1(t1)H3(t2)H1(t3)A1(t-t1-t2-t3)×A3(t-t2-t3)[A3(t-t3-τ)]*.
P(I)=S1(r)exp[-i(ω1t+ω2τ)]×-+dν w(ν)0dt30dt20dt1×exp[-iθI(ν)]H1(t1)H2(t2)×H1(t3)u1(t-t1-t2-t3)×u2(t-t2-t3)u2*(t-t3-τ).
P(II)=S2(r)exp[-i(ω1t+ω3τ-ω3δτ)]×-+dν w(ν)0dt30dt20dt1×exp[-iθII(ν)]H1(t1)H3(t2)×H1(t3)u1(t-t1-t2-t3)×u3(t-t2-t3)u3*×(t-t3-τ+δτ),
S1(r)=-iNμ12μ22312(2)* exp[i(k1+k2-k2)·r],
S2(r)=-iNμ12μ32313(3)* exp[i(k1+k3-k3)·r],
θI(ν)=ν·[k1(t1+t2+t3)+k2(t2+t3)-k2t3],
θII(ν)=ν·[k1(t1+t2+t3)+k3(t2+t3)-k3t3].
ui(t1)ui(t2)ui*(t3)ui*(t4)=ui(t1)ui*(t3)ui(t2)ui*(t4)+ui(t1)ui*(t4)ui(t2)ui*(t3).
ui(t1)ui*(t2)=exp(-αi|t1-t2|),(i=2, 3),
I(τ, r)|P(3)|2=B1+|η|2B2+|B3|2 exp (-2α2|τ|)+|ηB4|2 exp (-2α3|τ-δτ|)+exp(-α2|τ|-α3|τ-δτ|)×{ηB3*B4 exp[-iΔk·r-i(ω3-ω2)τ+iω3δτ]+η*B3B4*×exp[iΔk·r+i(ω3-ω2)τ-iω3δτ]},
Δk=(k2-k2)-(k3-k3),
B1=B3Γ10+2α2Γ10[Γ20-i(Δ1+Δ2)]-Γ102+Δ122Γ10Γ20(Γ10+Γ20-iΔ2),
B2=B4Γ10+2α3Γ10[Γ30-i(Δ1+Δ3)]-Γ102+Δ122Γ10Γ30(Γ10+Γ30-iΔ3),
B3=1Γ20+iΔ1+iΔ2,
B4=1Γ30+iΔ1+iΔ3,
η=μ323(3)*μ222(2)*.
I(τ, r)|P(3)|2=B5+|η|2B6+|B7|2 exp(-2Γ20|τ|)+|ηB8|2 exp(-2Γ30|τ-δτ|)+B7B8 exp(-Γ20|τ|-Γ30|τ-δτ|)×{η exp[-iΔk·r-i(Ω3-Ω2)τ+iΩ3δτ]+η* exp[iΔk·r+i(Ω3-Ω2)τ-iΩ3δτ]},
B5=Γ102+Δ122Γ102α22+iα2Δ1-iΓ20Δ2α2Γ20(2α2+iΔ1)[α22+(Δ1+Δ2)2]-1α2-i(Δ1+Δ2)×1(2α2-iΔ1)[α2-i(Δ1+Δ2)]-Γ10+Γ20+iΔ2Γ20[(Γ10+iΔ2)2-α22],
B6=Γ102+Δ122Γ102α32+iα3Δ1α3Γ30(2α3+iΔ1)[α32+(Δ1+Δ3)2]-1α3-i(Δ1+Δ3)×1(2α3-iΔ1)[α3-i(Δ1+Δ3)]-Γ10+Γ30+iΔ3Γ30[(Γ10+iΔ3)2-α32],
B7=2α2α22+(Δ1+Δ2)2,
B8=2α3α32+(Δ1+Δ3)2.
I(τ, r)|P(3)|2=B5+|η|2B6+|B7|2 exp(-2Γ20|τ|)+|ηB9|2 exp(-2α3|τ-δτ|)+B7 exp(-Γ20|τ|-α3|τ-δτ|)×{ηB9 exp[-iΔk·r-i(ω3-ω2)τ+iω3δτ-i(Δ1+Δ2)τ]+η*B9*exp[iΔk·r+i(ω3-ω2)τ-iω3δτ+i(Δ1+Δ2)τ]},
B9=1α3-i(Δ1+Δ2).
I(τ, r)|P(3)|2=B5+|η|2B6+|B10|2 exp(-2α2|τ|)+|ηB11|2 exp(-2α3|τ-δτ|)+exp(-α2|τ|-α3|τ-δτ|)×{ηB10*B11 exp[-iΔk·r-i(ω3-ω2)τ+iω3δτ]+η*B10B11* exp[iΔk·r+i(ω3-ω2)τ-iω3δτ]},
B10=1α2+i(Δ1+Δ2),B11=1α3+i(Δ1+Δ3).
-+dνw(ν)exp[-iθI(ν)]2πk1uδ(t1+t2+t3-ξ1t2),
-+dνw(ν)exp[-iθII(ν)]2πk1uδ(t1+t2+t3-ξ2t2),
I(τ, r)|P(3)|2=|P(I)+P(II)|2.
I(τ, r)|P(3)|2=B5+|η|2B6+|B12|2 exp(-2α2|τ|)+|ηB13|2 exp(-2α3|τ-δτ|)+exp(-α2|τ|-α3|τ-dτ|)×{ηB12*B13 exp[-iΔk·r-i(ω3-ω2)τ+iω3δτ]+η*B12B13* exp[iΔk·r+i(ω3-ω2)τ-iω3δτ]},
B12=(ξ1-1)(Γ102+Δ12)2(Γ20a-Γ10)2+(Δ2a)2,
B13=(ξ2-1)(Γ102+Δ12)2(Γ30a-Γ10)2+(Δ3a)2,
Γ20a=Γ20+ξ1Γ10,Γ30a=Γ30+ξ2Γ10,
Δ2a=Δ2+ξ1Δ1,Δ3a=Δ3+ξ2Δ1.
I(τ, r)|P(3)|2=B14+|η|2B15+|B16|2×exp[-2(Γ20a-Γ10)|τ|]+|ηB17|2×exp[-2(Γ30a-Γ10)|τ-δτ|]+B16B17 exp[-(Γ20a-Γ10)|τ|-(Γ30a-Γ10)|τ-δτ|]{η exp[-iΔk·r-i(Ω3-Ω2)τ-i(ξ2-ξ1)Δ1τ+iξ2Δ1δτ]+η* exp[iΔk·r+i(Ω3-Ω2)τ+i(ξ2-ξ1)Δ1τ-iξ2Δ1δτ]},
B14=(ξ1-1)[α22+(Δ2a)2-2iα2Δ2a]2(Γ20a-Γ10)2[α22+(Δ2a)2],
B15=(ξ2-1)[α32+(Δ3a)2-2iα3Δ3a]2(Γ30a-Γ10)2[α32+(Δ3a)2],
B16=2(ξ1-1)α2τα22+(Δ2a)2,
B17=2(ξ2-1)α3(τ-δτ)α32+(Δ3a)2.
I(τ, r)|P(3)|2=B14+|η|2B15+|B18|2 exp(-2α2|τ|)+|ηB19|2 exp(-2α3|τ-δτ|)+exp(-α2|τ|-α3|τ-δτ|){ηB18*B19 exp[-iΔk·r-i(ω3-ω2)τ+iω3δτ]+η*B18B19*×exp[iΔk·r+i(ω3-ω2)τ-iω3δτ]},
B18=ξ1-1(α2-iΔ2a)2,B19=ξ2-1(α3-iΔ3a)2.
I(τ, r)|P(3)|2=B14+|η|2B15+|B16|2×exp[-2(Γ20a-Γ10)|τ|]+|ηB19|2×exp(-2α3|τ-δτ|)+B16×exp[-2(Γ20a-Γ10)|τ|-α3|τ-δτ|]×{ηB19 exp[-iΔk·r-i(ω3-ω2)τ+iω3δτ-i(Δ1+Δ2)τ]+η*B19*exp[iΔk·r+i(ω3-ω2)τ-iω3δτ+i(Δ1+Δ2)τ]}.
ui(t1) ui*(t2)=exp(-αi|t1-t2|)2/αiδ(t1-t2),i=2, 3.
I(τ, r)|P(3)|2=A1+|η|2A2+|A3|2×exp[-2(Γ20a-Γ10)|τ|]+|ηA4|2×exp[-2(Γ30a-Γ10)|τ-δτ|]+A3A4×exp[-(Γ20a-Γ10)|τ|-(Γ30a-Γ10)|τ-δτ|]×{η exp[-iΔk·r-i(Ω3-Ω2)τ-i(ξ2-ξ1)Δ1τ+iξ2Δ1δτ]+η* exp[iΔk·r+i(Ω3-Ω2)τ+i(ξ2-ξ1)Δ1τ-iξ2Δ1δτ]},
A1=ξ1-14[α2(Γ20a-Γ10)2],A2=ξ2-14[α3(Γ30a-Γ10)]2,
A3=(ξ1-1)τα2,A4=(ξ2-1)(τ-δτ)α3,
I(τ, r)|P(3)|2=A1+|η|2A2+|A3|2 exp[-2(Γ20a-Γ10)|τ|].
ui(t1)ui(t2)ui*(t3)ui*(t4)=exp[-αi(|t1-t3|+|t1-t4|+|t2-t3|+|t2-t4|)]exp[αi(|t1-t2|+|t3-t4|)].
I(τ, r)|P(3)|2=|A3|2 exp[-2(Γ20a-Γ10)|τ|]+|ηA4|2 exp[-2(Γ30a-Γ10)|τ-δτ|]+A3A4×exp[-(Γ20a-Γ10)|τ|-(Γ30a-Γ10)|τ-δτ|]×{η exp[-iΔk·r-i(Ω3-Ω2)×τ-i(ξ2-ξ1)Δ1τ+iξ2Δ1δτ]+η* exp[iΔk·r+i(Ω3-Ω2)×τ+i(ξ2-ξ1)Δ1τ-iξ2Δ1δτ]}.
I(τ, r)|P(3)|2=|A3|2 exp[-2(Γ20a-Γ10)|τ|].
ui(t1)ui(t2)ui(t3)ui(t4)=ui(t1)ui(t3)ui(t2)ui(t4)+ui(t1)ui(t4)×ui(t2)ui(t3)+ui(t1)ui(t2)ui(t3)ui(t4).
I(τ, r)|P(3)|2=A1+|η|2A2+A5×exp[-2(Γ20a-Γ10)|τ|]+|ηA6|2×exp[-2(Γ30a-Γ10)|τ-δτ|]+A7×exp[-(Γ20a-Γ10)|τ|-(Γ30a-Γ10)|τ-δτ|]×{η exp[-iΔk·r-i(Ω3-Ω2)×τ-i(ξ2-ξ1)Δ1τ+iξ2Δ1δτ]+η* exp[iΔk·r+i(Ω3-Ω2)τ+i(ξ2-ξ1)Δ1τ-iξ2Δ1δτ]},
whereA5=4(ξ1-1)2τ2α22+ξ1-1(α2Δ2a)2,
A6=4(ξ2-1)2(τ-δτ)2α32+ξ2-1(α3Δ3a)2,
A7=(ξ1-1)(ξ2-1)α2α3τ(τ-δτ).
I(τ, r)|P(3)|2=A1+|η|2A2+A5×exp[-2(Γ20a-Γ10)|τ|].
ui(t)ui(t+τ1)ui*(t+τ2)ui*(t+τ3)=ui(t)ui*(t+τ2)ui(t+τi)ui*(t+τ3)+ui(t)ui*(t+τ3)ui(t+τ1)ui*(t+τ2)+f(τ1, τ2, τ3),i=2,3,
S1(ω)=2α2/π4α22+(ω-ω2)2
S2(ω)=2α3/π4α32+(ω-ω3)2

Metrics