Abstract

We calculate the transient degenerate four-wave mixing signal with ultrashort laser pulses within the self-consistent formulation of the quantum kinetics of electrons and holes in a semiconductor interacting with longitudinal-optical (LO) phonons. Our numerical calculation shows that the integrated four-wave mixing signal as a function of the delay time shows an effective exponential decay, which, however, is modulated by an oscillation with a characteristic energy given by ωLO[1 + (me/mh)]. This peculiar quantum beat is caused by the interference of the interband polarization components, which are connected through a LO-phonon transition in the conduction band.

© 1996 Optical Society of America

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  1. P. Lipavský, V. Špička, and B. Velický, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933–6942 (1986).
    [CrossRef]
  2. A. P. Jauho, in Granular Nanoelectronics, D. K. Ferry, ed. (Plenum, New York, 1991), p. 133.
    [CrossRef]
  3. A. V. Kuznetsov, “Interaction of ultrashort light pulses with semiconductors: effective Bloch equations with relaxation and memory effect,” Phys. Rev. B 44, 8721–8744 (1991).
    [CrossRef]
  4. H. Haug and A. P. Jauho, Quantum Kinetics for Transport and Optics in Semiconductors (Springer, Berlin, 1996).
  5. H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse-excited semiconductor,” Phys. Status Solidi B 173, 139–142 (1992).
    [CrossRef]
  6. L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
    [CrossRef]
  7. K. El Sayed, “Modellierung ultraschneller Prozesse in optisch angeregten Halbleiterplasmen,” Ph.D. dissertation (Institut für Theoretische Physik, J. W. Goethe Universität Frankfurt, Frankfurt am Main, Germany, 1994).
  8. D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
    [CrossRef]
  9. D. B. Tran Thoai and H. Haug, “Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors,” Phys. Rev. B 47, 3574–3581 (1993).
    [CrossRef]
  10. J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435–5447 (1994).
    [CrossRef]
  11. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 3rd ed. (World Scientific, Singapore, 1994), Chap. 12, pp. 218–224.
    [CrossRef]
  12. K. El Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. A 50, 1541–1549 (1994).
  13. M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon echo,” Phys. Rev. A 45, 1865–1875 (1992).
    [CrossRef] [PubMed]
  14. L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
    [CrossRef]

1995 (2)

D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
[CrossRef]

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

1994 (2)

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435–5447 (1994).
[CrossRef]

K. El Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. A 50, 1541–1549 (1994).

1993 (1)

D. B. Tran Thoai and H. Haug, “Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors,” Phys. Rev. B 47, 3574–3581 (1993).
[CrossRef]

1992 (3)

M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon echo,” Phys. Rev. A 45, 1865–1875 (1992).
[CrossRef] [PubMed]

H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse-excited semiconductor,” Phys. Status Solidi B 173, 139–142 (1992).
[CrossRef]

L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
[CrossRef]

1991 (1)

A. V. Kuznetsov, “Interaction of ultrashort light pulses with semiconductors: effective Bloch equations with relaxation and memory effect,” Phys. Rev. B 44, 8721–8744 (1991).
[CrossRef]

1986 (1)

P. Lipavský, V. Špička, and B. Velický, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933–6942 (1986).
[CrossRef]

Bányai, L.

D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
[CrossRef]

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

K. El Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. A 50, 1541–1549 (1994).

L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
[CrossRef]

Binder, R.

M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon echo,” Phys. Rev. A 45, 1865–1875 (1992).
[CrossRef] [PubMed]

El Sayed, K.

K. El Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. A 50, 1541–1549 (1994).

K. El Sayed, “Modellierung ultraschneller Prozesse in optisch angeregten Halbleiterplasmen,” Ph.D. dissertation (Institut für Theoretische Physik, J. W. Goethe Universität Frankfurt, Frankfurt am Main, Germany, 1994).

Haug, H.

D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
[CrossRef]

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

K. El Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. A 50, 1541–1549 (1994).

D. B. Tran Thoai and H. Haug, “Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors,” Phys. Rev. B 47, 3574–3581 (1993).
[CrossRef]

L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
[CrossRef]

H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse-excited semiconductor,” Phys. Status Solidi B 173, 139–142 (1992).
[CrossRef]

H. Haug and A. P. Jauho, Quantum Kinetics for Transport and Optics in Semiconductors (Springer, Berlin, 1996).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 3rd ed. (World Scientific, Singapore, 1994), Chap. 12, pp. 218–224.
[CrossRef]

Jauho, A. P.

H. Haug and A. P. Jauho, Quantum Kinetics for Transport and Optics in Semiconductors (Springer, Berlin, 1996).

A. P. Jauho, in Granular Nanoelectronics, D. K. Ferry, ed. (Plenum, New York, 1991), p. 133.
[CrossRef]

Koch, S. W.

M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon echo,” Phys. Rev. A 45, 1865–1875 (1992).
[CrossRef] [PubMed]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 3rd ed. (World Scientific, Singapore, 1994), Chap. 12, pp. 218–224.
[CrossRef]

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, 5435–5447 (1994).
[CrossRef]

Kuznetsov, A. V.

A. V. Kuznetsov, “Interaction of ultrashort light pulses with semiconductors: effective Bloch equations with relaxation and memory effect,” Phys. Rev. B 44, 8721–8744 (1991).
[CrossRef]

Lindberg, M.

M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon echo,” Phys. Rev. A 45, 1865–1875 (1992).
[CrossRef] [PubMed]

Lipavský, P.

P. Lipavský, V. Špička, and B. Velický, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933–6942 (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, 5435–5447 (1994).
[CrossRef]

Reitsamer, E.

D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
[CrossRef]

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

Remling, C.

L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
[CrossRef]

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, 5435–5447 (1994).
[CrossRef]

Špicka, V.

P. Lipavský, V. Špička, and B. Velický, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933–6942 (1986).
[CrossRef]

Steinbach, D.

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

Tran Thoai, D. B.

D. B. Tran Thoai and H. Haug, “Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors,” Phys. Rev. B 47, 3574–3581 (1993).
[CrossRef]

TranThoai, D. B.

D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
[CrossRef]

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
[CrossRef]

Velický, B.

P. Lipavský, V. Špička, and B. Velický, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933–6942 (1986).
[CrossRef]

Wegener, M.

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

Wehner, M. U.

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

Phys. Rev. A (2)

K. El Sayed, L. Bányai, and H. Haug, “Coulomb quantum kinetics and optical dephasing on the femtosecond time scale,” Phys. Rev. A 50, 1541–1549 (1994).

M. Lindberg, R. Binder, and S. W. Koch, “Theory of the semiconductor photon echo,” Phys. Rev. A 45, 1865–1875 (1992).
[CrossRef] [PubMed]

Phys. Rev. B (4)

P. Lipavský, V. Špička, and B. Velický, “Generalized Kadanoff–Baym ansatz for deriving quantum transport equations,” Phys. Rev. B 34, 6933–6942 (1986).
[CrossRef]

A. V. Kuznetsov, “Interaction of ultrashort light pulses with semiconductors: effective Bloch equations with relaxation and memory effect,” Phys. Rev. B 44, 8721–8744 (1991).
[CrossRef]

D. B. Tran Thoai and H. Haug, “Band-edge quantum kinetics for coherent ultrashort-pulse spectroscopy in polar semiconductors,” Phys. Rev. B 47, 3574–3581 (1993).
[CrossRef]

J. Schilp, T. Kuhn, and G. Mahler, “Electron–phonon quantum kinetics in pulse excited semiconductors: memory and renormalization effects,” Phys. Rev. B 50, 5435–5447 (1994).
[CrossRef]

Phys. Rev. Lett. B (1)

L. Bányai, D. B. TranThoai, E. Reitsamer, H. Haug, D. Steinbach, M. U. Wehner, and M. Wegener, “Exciton–LO-phonon quantum kinetics: evidence of memory effects in bulk GaAs,” Phys. Rev. Lett. B 75, 2188–2190 (1995).
[CrossRef]

Phys. Status Solidi B (3)

D. B. TranThoai, L. Bányai, E. Reitsamer, and H. Haug, “Coherent interband effects in quantum kinetics,” Phys. Status Solidi B 188, 387–394 (1995).
[CrossRef]

H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse-excited semiconductor,” Phys. Status Solidi B 173, 139–142 (1992).
[CrossRef]

L. Bányai, D. B. TranThoai, C. Remling, and H. Haug, “Interband quantum kinetics with LO-phonon scattering in a laser-pulse excited semiconductor,” Phys. Status Solidi B 173, 149–158 (1992).
[CrossRef]

Other (4)

K. El Sayed, “Modellierung ultraschneller Prozesse in optisch angeregten Halbleiterplasmen,” Ph.D. dissertation (Institut für Theoretische Physik, J. W. Goethe Universität Frankfurt, Frankfurt am Main, Germany, 1994).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 3rd ed. (World Scientific, Singapore, 1994), Chap. 12, pp. 218–224.
[CrossRef]

H. Haug and A. P. Jauho, Quantum Kinetics for Transport and Optics in Semiconductors (Springer, Berlin, 1996).

A. P. Jauho, in Granular Nanoelectronics, D. K. Ferry, ed. (Plenum, New York, 1991), p. 133.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Calculated FWM signal | P 2 ( t , τ ) | as a function of the real time t for different delay times τ.

Fig. 2
Fig. 2

Calculated FWM signal Re P 2 ( t , τ ) as a function of the real time t for different delay times τ.

Fig. 3
Fig. 3

Calculated FWM signal Im P 2 ( t , τ ) as a function of the real time t for different delay times τ.

Fig. 4
Fig. 4

Calculated integrated FWM signal 0 d t | P 2 ( t , τ ) | 2 function of the delay time.

Fig. 5
Fig. 5

Dependence of the quantum beat frequency on the electron–hole mass ratio.

Equations (38)

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i G μ v , k ( t 1 , t 2 ) = σ G μ σ , k r ( t 1 , t 2 ) G σ v , k ( t 2 , t 2 ) G μ σ , k ( t 1 , t 1 ) G σ v , k a ( t 1 , t 2 ) .
G μ v , k < ( t 1 , t 2 ) = i a v , k ( t 2 ) a μ , k ( t 1 ) , G μ v , k > ( t 1 , t 2 ) = i a μ , k ( t 1 ) a v , k ( t 2 ) ,
G μ v , k < ( t , t ) = i ρ μ v , k ( t ) = i a v , k ( t ) a μ , k ( t ) , G μ v , k > ( t , t ) = i [ δ μ v ρ μ v , k ( t ) ] .
ρ c c , k ( t ) = f e , k ( t ) , ρ v v , k ( t ) = 1 f h , k ( t ) , ρ c v , k ( t ) = P k ( t ) exp ( i ω t ) ,
G μ v , k r ( t 1 , t 2 ) = i θ ( t 1 , t 2 ) [ a μ , k ( t 1 ) , a v , k ( t 2 ) ] L
G μ v , k r ( t 1 , t 2 ) = i θ ( t 1 , t 2 ) δ μ , v exp ( i u , k 0 γ μ , k ) ( t 1 , t 2 ) .
G μ v , k r ( t 2 , t 2 ) = i δ μ , v , G μ v , k < ( t 2 , t 2 ) = i ρ μ v , k ( t 2 ) .
G μ v , k < ( t 1 , t 2 ) = i G μ σ , k r ( t 1 , t 2 ) G σ v , k < ( t 2 , t 2 ) ,
E ( t ) = E 0 ( t ) exp ( i ω t ) ,
[ t + i Δ k ( t ) ] P k ( t ) = i 2 Ω R , k ( t ) [ 1 f h , k ( t ) f e , k ( t ) ] t P k ( t ) | scatt ,
t f i , k ( t ) = Im [ Ω R , k * ( t ) P k ( t ) ] t f i , k ( t ) | scatt .
Δ k ( t ) = e , k ( t ) + h , k ( t ) ω ,
i , k ( t ) = i , k 0 k V k k f i , k ( t ) ,
Ω R , k ( t ) = d k E 0 ( t ) + 2 k V k k P k ( t ) .
d k = d 0 exp ( e , k 0 + h , k 0 E g e 0 ) / δ 0 + 1 .
ρ μ v , k t | scatt = q , σ τ , ± ( g q 2 t d t { G μ σ , k q r ( t , t ) G τ v , k a ( t , t ) exp [ ± ω LO ( t , t ) ] × [ N q ± ρ σ τ , k q ( t ) N q ± ρ σ τ , k q ( t ) ± ρ ρ σ ρ , k q ( t ) ρ ρ τ , k ( t ) ] } { k k q } ) ,
g q 2 = α 4 π ( ω 0 ) 3 / 2 ( 2 μ ) 1 / 2 q 2 V , α = e 2 ( μ 2 ω 0 ) 1 / 2 ( 1 1 0 ) ,
N q ± = 1 exp ( β ω 0 1 ) + 1 2 ± 1 2 N q + { 1 0 } ,
H = μ , k μ , k ( t ) a μ , k k 1 2 [ Ω k ( t ) a c , k a v , k + h . c . ] .
a μ , k ( t ) = v U μ v , k ( t ) a v , k .
H k ( t ) U k ( t ) = i ћ d U k ( t ) d t ,
H k = [ c , k ( t ) 1 2 Ω k ( t ) 1 2 Ω k * ( t ) v , k ( t ) ] .
U μ v ( ) = δ μ v .
G μ v , k r ( t , t ) = i Θ ( t t ) [ a μ , k ( t ) , a v , k ( t ) ] + = i Θ ( t t ) [ U k ( t ) U k ( t ) ] μ v .
μ , k 0 μ , k 0 i Γ μ , k .
Γ μ , k = 2 π k g q 2 [ ( N q + 1 ) δ ( e μ , k ћ ω LO e μ , k q ) + N q δ ( e μ , k + ћ ω LO e μ , k q ) ] .
G μ v , k r ( t , t ) = i Θ ( t t ) [ U Γ , k ( t ) U Γ , k ( t ) ] μ v .
E ( t ) = E ( t ) exp [ i ( k 1 · x ω t ) ] + E ( t τ ) exp [ i ( k 2 · x ω t ) ] .
E ( t ) = E 0 sech [ ( t / δ t ) ] .
k p k ( t ) = exp ( i ϕ 1 ) n = P n exp [ i ( ϕ 2 ϕ 1 ) n ] .
E ( t ) = E ( t ) exp ( i ω t ) + E ( t τ ) exp [ i ( ϕ ω ) t ]
P 2 ( t , τ ) = 1 2 π 0 2 π d ϕ exp ( i 2 ϕ ) k p k ( t , τ , ϕ ) ,
P 2 ( t , τ ) = lim M 1 M m = 0 M exp [ i ( 4 m π / M ) ] k p k ( t , τ ; 2 m π M ) .
0 d t | P 2 ( t , τ ) | 2 = 0 d ω | P ˜ 2 ( ω , τ ) | 2 ,
0 d t | P 2 ( t , τ ) | 2 [ 1 + a sin ( ω qb τ α ) ] exp [ ( t / T eff ) ] ,
ω qb = ω LO ( 1 + m e m h ) .
ћ 2 k 2 2 m e ћ 2 k 2 2 m e = ћ ω LO ,
ћ 2 k 2 2 μ ћ 2 k 2 2 μ = ћ ω LO ( 1 + m e m h ) .

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