Abstract

The influence of electron–electron scattering processes on the time evolution of four-wave mixing signals for ultrashort excitation in the vicinity of the band edge is investigated. The validity of a Markovian approximation is carefully analyzed. The Markovian relaxation of the one-particle distribution is compared with non-Markovian results obtained from the solution of quantum-kinetic equations for two-point functions. In the non-Markovian case relaxation processes are slowed down.

© 1996 Optical Society of America

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References

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  1. D. B. Tran Thoai and H. Haug, “Coulomb quantum kinetics in pulse-excited semiconductors,” Z. Phys. B 91, 199 (1993).
    [CrossRef]
  2. M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).
  3. K. El Sayed, L. Banyai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond timescale,” Phys. Rev. B 50, 1541 (1994).
    [CrossRef]
  4. K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
    [CrossRef]
  5. L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
    [CrossRef]
  6. M. Hartmann and W. Schäfer, “Real time approach to relaxation and dephasing processes in semiconductors,” Phys. Status Solidi B 173, 165 (1992).
    [CrossRef]
  7. R. Zimmermann, “Transverse relaxation and polarization specifics in the dynamical Stark effect,” Phys. Status Solidi B 159, 317 (1990).
    [CrossRef]
  8. J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse-excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435 (1994).
    [CrossRef]
  9. W. Schäfer, “Many-body effects in nonlinear optics of semiconductor structures,” in Optics of Semiconductor Nano-structures, F. Henneberger, S. Schmitt-Rink, and E. O. Göbel, eds. (Akademie-Verlag, Berlin, 1993), p. 21, and references therein.
  10. Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
    [CrossRef]
  11. T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
    [CrossRef]
  12. K. Morawetz and G. Röpke, “Memory effects and virial corrections in nonequilibrium dense systems,” Phys. Rev. E 51, 4246 (1995).
    [CrossRef]
  13. P. Danielewicz, “Quantum theory of nonequilibrium processes, I,” Ann. Phys. 152, 305 (1984).
    [CrossRef]
  14. P. Lipavsky, V. Spicka, and B. Velicky, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933 (1986).
    [CrossRef]
  15. M. U. Wehner, D. Steinbach, M. Wegener, T. Marschner, and W. Stolz, “Spectrally resolved four-wave mixing experiments on bulk GaAs with 14-fs pulses,” J. Opt. Soc. Am. B 13, 977–980 (1996).
    [CrossRef]
  16. H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
    [CrossRef] [PubMed]
  17. P. Danielewicz, “Quantum theory of nonequilibrium processes, II,” Ann. Phys. 152, 239 (1984).
    [CrossRef]

1996 (1)

1995 (1)

K. Morawetz and G. Röpke, “Memory effects and virial corrections in nonequilibrium dense systems,” Phys. Rev. E 51, 4246 (1995).
[CrossRef]

1994 (5)

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse-excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435 (1994).
[CrossRef]

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
[CrossRef]

K. El Sayed, L. Banyai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond timescale,” Phys. Rev. B 50, 1541 (1994).
[CrossRef]

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

1993 (2)

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

D. B. Tran Thoai and H. Haug, “Coulomb quantum kinetics in pulse-excited semiconductors,” Z. Phys. B 91, 199 (1993).
[CrossRef]

1992 (2)

L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
[CrossRef]

M. Hartmann and W. Schäfer, “Real time approach to relaxation and dephasing processes in semiconductors,” Phys. Status Solidi B 173, 165 (1992).
[CrossRef]

1990 (1)

R. Zimmermann, “Transverse relaxation and polarization specifics in the dynamical Stark effect,” Phys. Status Solidi B 159, 317 (1990).
[CrossRef]

1986 (1)

P. Lipavsky, V. Spicka, and B. Velicky, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933 (1986).
[CrossRef]

1984 (2)

P. Danielewicz, “Quantum theory of nonequilibrium processes, II,” Ann. Phys. 152, 239 (1984).
[CrossRef]

P. Danielewicz, “Quantum theory of nonequilibrium processes, I,” Ann. Phys. 152, 305 (1984).
[CrossRef]

Banyai, L.

K. El Sayed, L. Banyai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond timescale,” Phys. Rev. B 50, 1541 (1994).
[CrossRef]

L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
[CrossRef]

Binder, R.

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

Bonitz, M.

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

Cundiff, S. T.

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

Danielewicz, P.

P. Danielewicz, “Quantum theory of nonequilibrium processes, II,” Ann. Phys. 152, 239 (1984).
[CrossRef]

P. Danielewicz, “Quantum theory of nonequilibrium processes, I,” Ann. Phys. 152, 305 (1984).
[CrossRef]

El Sayed, K.

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

K. El Sayed, L. Banyai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond timescale,” Phys. Rev. B 50, 1541 (1994).
[CrossRef]

Ferrio, K. B.

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

Hartmann, M.

M. Hartmann and W. Schäfer, “Real time approach to relaxation and dephasing processes in semiconductors,” Phys. Status Solidi B 173, 165 (1992).
[CrossRef]

Haug, H.

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

K. El Sayed, L. Banyai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond timescale,” Phys. Rev. B 50, 1541 (1994).
[CrossRef]

D. B. Tran Thoai and H. Haug, “Coulomb quantum kinetics in pulse-excited semiconductors,” Z. Phys. B 91, 199 (1993).
[CrossRef]

L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
[CrossRef]

Henneberger, K.

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

Herzel, F.

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

Hu, Y. Z.

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

Koch, S. W.

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

Kock, S. W.

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

Köhler, H. S.

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

Kraeft, W. D.

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

Kremp, D.

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

Kuhn, T.

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse-excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435 (1994).
[CrossRef]

Lipavsky, P.

P. Lipavsky, V. Spicka, and B. Velicky, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933 (1986).
[CrossRef]

Mahler, G.

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse-excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435 (1994).
[CrossRef]

Marschner, T.

Morawetz, K.

K. Morawetz and G. Röpke, “Memory effects and virial corrections in nonequilibrium dense systems,” Phys. Rev. E 51, 4246 (1995).
[CrossRef]

Peter, U.

T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
[CrossRef]

Rappen, T.

T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
[CrossRef]

Remling, C.

L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
[CrossRef]

Röpke, G.

K. Morawetz and G. Röpke, “Memory effects and virial corrections in nonequilibrium dense systems,” Phys. Rev. E 51, 4246 (1995).
[CrossRef]

Schäfer, W.

T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
[CrossRef]

M. Hartmann and W. Schäfer, “Real time approach to relaxation and dephasing processes in semiconductors,” Phys. Status Solidi B 173, 165 (1992).
[CrossRef]

W. Schäfer, “Many-body effects in nonlinear optics of semiconductor structures,” in Optics of Semiconductor Nano-structures, F. Henneberger, S. Schmitt-Rink, and E. O. Göbel, eds. (Akademie-Verlag, Berlin, 1993), p. 21, and references therein.

Schilp, J.

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse-excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435 (1994).
[CrossRef]

Schuster, S.

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

Scott, D. C.

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

Spicka, V.

P. Lipavsky, V. Spicka, and B. Velicky, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933 (1986).
[CrossRef]

Steel, D. G.

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

Steinbach, D.

Stolz, W.

Tran Thoai, D. B.

D. B. Tran Thoai and H. Haug, “Coulomb quantum kinetics in pulse-excited semiconductors,” Z. Phys. B 91, 199 (1993).
[CrossRef]

L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
[CrossRef]

Velicky, B.

P. Lipavsky, V. Spicka, and B. Velicky, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933 (1986).
[CrossRef]

Wang, H.

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

Wegener, M.

M. U. Wehner, D. Steinbach, M. Wegener, T. Marschner, and W. Stolz, “Spectrally resolved four-wave mixing experiments on bulk GaAs with 14-fs pulses,” J. Opt. Soc. Am. B 13, 977–980 (1996).
[CrossRef]

T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
[CrossRef]

Wehner, M. U.

Zimmermann, R.

R. Zimmermann, “Transverse relaxation and polarization specifics in the dynamical Stark effect,” Phys. Status Solidi B 159, 317 (1990).
[CrossRef]

Ann. Phys. (2)

P. Danielewicz, “Quantum theory of nonequilibrium processes, I,” Ann. Phys. 152, 305 (1984).
[CrossRef]

P. Danielewicz, “Quantum theory of nonequilibrium processes, II,” Ann. Phys. 152, 239 (1984).
[CrossRef]

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

Phys. Rev. B (6)

P. Lipavsky, V. Spicka, and B. Velicky, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933 (1986).
[CrossRef]

Y. Z. Hu, R. Binder, S. W. Kock, S. T. Cundiff, H. Wang, and D. G. Steel, “Excitation and polarization effects in semiconductor four-wave-mixing spectroscopy,” Phys. Rev. B 49, 14,382 (1994).
[CrossRef]

T. Rappen, U. Peter, M. Wegener, and W. Schäfer, “Polarization dependence of dephasing processes: a probe for many-body effects,” Phys. Rev. B 49, 10774 (1994).
[CrossRef]

K. El Sayed, L. Banyai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond timescale,” Phys. Rev. B 50, 1541 (1994).
[CrossRef]

K. El Sayed, S. Schuster, H. Haug, F. Herzel, and K. Henneberger, “Subpicosecond plasmon response: buildup of screening,” Phys. Rev. B 49, 7337 (1994).
[CrossRef]

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse-excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435 (1994).
[CrossRef]

Phys. Rev. E (1)

K. Morawetz and G. Röpke, “Memory effects and virial corrections in nonequilibrium dense systems,” Phys. Rev. E 51, 4246 (1995).
[CrossRef]

Phys. Rev. Lett. (1)

H. Wang, K. B. Ferrio, D. G. Steel, Y. Z. Hu, R. Binder, and S. W. Koch, “Transient nonlinear optical response from excitation induced dephasing in GaAs,” Phys. Rev. Lett. 71, 1261 (1993).
[CrossRef] [PubMed]

Phys. Status Solidi B (3)

L. Banyai, D. B. Tran Thoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149 (1992).
[CrossRef]

M. Hartmann and W. Schäfer, “Real time approach to relaxation and dephasing processes in semiconductors,” Phys. Status Solidi B 173, 165 (1992).
[CrossRef]

R. Zimmermann, “Transverse relaxation and polarization specifics in the dynamical Stark effect,” Phys. Status Solidi B 159, 317 (1990).
[CrossRef]

Z. Phys. B (1)

D. B. Tran Thoai and H. Haug, “Coulomb quantum kinetics in pulse-excited semiconductors,” Z. Phys. B 91, 199 (1993).
[CrossRef]

Other (2)

M. Bonitz, D. C. Scott, R. Binder, D. Kremp, W. D. Kraeft, and H. S. Köhler, “Memory effects in two-particle collisions,” in Proceedings of the International Conference on the Physics of Strongly Coupled Plasmas, W. D. Kraeft and M. Schlanges, eds. (World Scientific, Singapore, 1996).

W. Schäfer, “Many-body effects in nonlinear optics of semiconductor structures,” in Optics of Semiconductor Nano-structures, F. Henneberger, S. Schmitt-Rink, and E. O. Göbel, eds. (Akademie-Verlag, Berlin, 1993), p. 21, and references therein.

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Figures (7)

Fig. 1
Fig. 1

Frequency dependence of the integrand in Eq. (2.13) and of positive- and negative-frequency parts for t = 30 fs and different k values: k = a B 1, solid curves; k = 4 a B 1, dotted–dashed curves; k = 8 a B 1, dashed curves. Starting from the left, the uppermost (lowest) group of curves describes the positive (negative) frequency contributions; the sum of both is given by the intermediate group of curves.

Fig. 2
Fig. 2

Time and wave-vector dependence of the electron (E) distribution function (F) on a scale from 0 to 0.5. The k axes range from 0 to 8 a B 1 (from left to right); the T axes, from −0.3 to 1.2 ps (from front to back).

Fig. 3
Fig. 3

Time and wave-vector dependence of the polarization function. The T axes range from −0.3 to 1.2 ps (from left to right); the k axes, from 0 to 8 a B 1 (from back to front).

Fig. 4
Fig. 4

TI FWM signals for pulse durations and Rabi frequencies of 18 fs and 0.15 Ry, 36 fs and 0.15 Ry, and 36 fs and 0.45 Ry.

Fig. 5
Fig. 5

TR FWM signals for pulse durations and Rabi frequencies of 18 fs and 0.15 Ry and different time delays: 0, 200, 300, and 400 fs (from top to bottom).

Fig. 6
Fig. 6

SR FWM signals for pulse durations and Rabi frequencies of 18 fs and 0.15 Ry and different time delays: 0, 200, 300, and 400 fs (second from top to bottom). The uppermost curve is the spectral shape of the excitation pulse.

Fig. 7
Fig. 7

Markovian versus non-Markovian relaxation of the one-particle distribution function for a pulse duration of 100 fs and a Rabi frequency of 0.8 Ry. The times are 50, 250, 450, and 600 fs in the non-Markovian case (bottom to top).

Equations (17)

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

[ i t n ( k ) + m ( k ) ] G n m ( k , t , t ) = l { [ d n l ( k ) E ( t ) + Σ n l s t ( k , t ) ] G l m ( k , t , t ) + d t [ Σ n l r ( k , t , t ) G l m ( k , t , t ) + Σ n l ( k , t , t ) G l m a ( k , t , t ) ] }
[ i t n ( k ) + m ( k ) ] G n m ( k , t , t ) = l { G n l ( k , t , t ) [ d l m ( k ) E ( t ) + Σ l m s t ( k , t ) ] + d t [ G n l r ( k , t , t ) Σ l m < ( k , t , t ) + G n l ( k , t , t ) Σ l m a ( k , t , t ) ] } .
G n m r ( k , t , t ) = θ ( t t ) [ G n m < ( k , t , t ) G n m > ( k , t , t ) ] ,
G n m a ( k , t , t ) = θ ( t t ) [ G n m > ( k , t , t ) G n m < ( k , t , t ) ] .
G n m ( k t , t ) = exp { i [ n ( k ) t m ( k ) t ] } g n m ( k , t , t ) ,
i t g n m < ( k , t , t τ ) = l ( Δ n l ( t ) g l m < ( k , t , t τ ) g n l < ( k , t , t τ ) Δ l m ( k , t τ ) + t τ d t [ σ n l > ( k , t , t ) g l m < ( k , t , t τ ) σ n l < ( k , t , t ) g l m > ( k , t , t τ ) + g n l < ( k , t , t ) σ l m > ( k , t , t τ ) g n l > ( k , t , t ) σ l m < ( k , t , t τ ) ] + t τ t d t { [ σ n l > ( k , t , t ) σ n l < ( k , t , t ) ] g l m < ( k , t , t τ ) [ g n l > ( k , t , t ) g n l < ( k , t , t ) ] σ l m < ( k , t , t τ ) } ) ,
Δ n m ( k , t ) = F n m ( t ) + σ n m s t ( k , t ) ,
F n m ( t ) = exp { i [ n ( k ) m ( k ) ] t } d n m ( k ) E ( t )
σ n m s t ( k , t ) = i k v ( k ) ( exp { i [ n ( k ) n ( k ) + m ( k ) m ( k ) ] t } × [ g n m < ( k + k , t , t ) δ n v δ n m ] ) ,
σ n m < ( k t t ) = k k n m ( w ( k , t ) w ( k , t ) g n m < ( k + k , t t ) g n m > × ( k + k , t t ) g m n < ( k , t t ) × exp { i [ ( k + k n k n k + k m + k m ) t ( k + k m k m k + k n + k n ) t ] } w ( k , t ) w ( k k , t ) g n n < ( k + k , t t ) g n m > × ( k + k , t t ) g m m < ( k , t t ) × exp { i [ ( k + k n k n k + k m + k m ) t ( k + k n k n k + k m + k m ) t ] } ) .
S c c ( k t t ) = k k t d t w ( k , t ) w ( k , t ) × { [ g c c < ( k + k , t t ) g c c > ( k + k , t t ) × g c c < ( k , t t ) g c c > ( k , t t ) ] ( > < , < > ) } cos [ ( k + k c k c k + k c + k c ) ( t t ) ] .
cos ( x t ) = d ω cos ( ω t ) δ ( x ω ) = 0 d ω cos ( ω t ) [ δ ( x ω ) + δ ( x + ω ) ] ,
S c c ( k t t ) = t d t 0 d ω k k w ( k , t ) w ( k , t ) × { [ g c c < ( k + k , t t ) g c c > ( k + k , t t ) × g c c < ( k , t t ) g c c > ( k , t t ) ] ( > < , < > ) } [ δ ( k + k c k c k + k c + k c + ω ) + δ ( k + k c k c k + k c + k c ω ) ] cos [ ω ( t t ) ] = t d t 0 d ω I c c ( k t t ω ) cos [ ω ( t t ) ] .
g n m < ( t , t ) = i l g n l r ( t t ) g l m < ( t t ) g n l < ( t t ) g l m a ( t t ) ,
P 2 q 2 q 1 ( t , T ) = k d v c P c v 2 q 2 q 1 ( k , t , T ) ,
S TI ( T ) = d t | P 2 q 2 q 1 ( t , T ) | 2 .
S SR ( ω , T ) = | P ( ω , T ) | 2 .

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