Abstract

In a self-consistent field formalism we investigated the weak-probe optical intersubband response of a three-subband semiconductor quantum well pumped by a strong coherent laser beam. We showed that, when the pumping photon energy is somewhat above the energy separation between the two lowest subbands, a combination of hole-burning effects and coherent pump–probe wave interactions can lead to both optical gain without population inversion and an Autler–Townes-like doublet in the probe optical spectrum. These two absorption peaks, whose positions are blueshifted as the pump intensity is increased, are due to the two-photon absorption process and light-induced intersubband transitions between the two upper states. It is also demonstrated that, in a stepwise two-photon pumping scheme, the optical gain with population inversion can also occur in the quantum-well system when a modestly strong pump field is present.

© 1998 Optical Society of America

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References

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  1. Y. Zhao, D. Huang, and C. Wu, Opt. Lett. 19, 816 (1994).
    [CrossRef] [PubMed]
  2. D. Huang, C. Wu, and Y. Zhao, J. Opt. Soc. Am. B 11, 2258 (1994).
    [CrossRef]
  3. D. S. Lee and K. J. Malloy, IEEE J. Quantum Electron. QE-30, 85 (1994).
    [CrossRef]
  4. D. Huang and Y. Zhao, Phys. Rev. A 51, 1617 (1995).
    [CrossRef] [PubMed]
  5. Y. Zhao, D. Huang, and C. Wu, J. Nonlin. Opt. Phys. Mater. 4, 261 (1995).
    [CrossRef]
  6. J. B. Khurgin and E. Rosencher, IEEE J. Quantum Electron. QE-32, 1882 (1996).
    [CrossRef]
  7. S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
    [CrossRef] [PubMed]
  8. A. Imamoglu and R. J. Ram, Opt. Lett. 19, 1744 (1994).
    [CrossRef]
  9. Y. Zhao, D. Huang, and C. Wu, J. Opt. Soc. Am. B 13, 1614 (1996).
    [CrossRef]
  10. G. S. Agarwal, Phys. Rev. A 51, R2711 (1995).
    [CrossRef]
  11. A. Liu, Phys. Rev. A 56, 3206 (1997).
    [CrossRef]
  12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  13. G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Paris, 1988).
  14. H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).
  15. Since the local-field (depolarization) effect is included in our formalism, it is expected that inclusion of the conduction band nonparabolicity only results in a slight change in our final results [M. Zaluzny, Phys. Rev. B 43, 4511 (1991)].
    [CrossRef]
  16. A. Liu and O. Keller, Phys. Rev. A 177, 441 (1993).
  17. J. E. Sipe, J. Opt. Soc. Am. B 4, 481 (1987), and references therein.
    [CrossRef]
  18. O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).
    [CrossRef]
  19. A. Liu, Phys. Rev. B 50, 8569 (1994).
    [CrossRef]
  20. M. Zaluzny, Phys. Rev. B 47, 3995 (1993).
    [CrossRef]
  21. A. Liu and O. Keller, Phys. Scr. 52, 116 (1995).
    [CrossRef]
  22. J. B. Khurgin, G. Sun, L. R. Friedman, and R. A. Soref, J. Appl. Phys. 78, 7398 (1995).
    [CrossRef]

1997 (1)

A. Liu, Phys. Rev. A 56, 3206 (1997).
[CrossRef]

1996 (2)

J. B. Khurgin and E. Rosencher, IEEE J. Quantum Electron. QE-32, 1882 (1996).
[CrossRef]

Y. Zhao, D. Huang, and C. Wu, J. Opt. Soc. Am. B 13, 1614 (1996).
[CrossRef]

1995 (5)

D. Huang and Y. Zhao, Phys. Rev. A 51, 1617 (1995).
[CrossRef] [PubMed]

Y. Zhao, D. Huang, and C. Wu, J. Nonlin. Opt. Phys. Mater. 4, 261 (1995).
[CrossRef]

G. S. Agarwal, Phys. Rev. A 51, R2711 (1995).
[CrossRef]

A. Liu and O. Keller, Phys. Scr. 52, 116 (1995).
[CrossRef]

J. B. Khurgin, G. Sun, L. R. Friedman, and R. A. Soref, J. Appl. Phys. 78, 7398 (1995).
[CrossRef]

1994 (5)

1993 (2)

M. Zaluzny, Phys. Rev. B 47, 3995 (1993).
[CrossRef]

A. Liu and O. Keller, Phys. Rev. A 177, 441 (1993).

1991 (1)

Since the local-field (depolarization) effect is included in our formalism, it is expected that inclusion of the conduction band nonparabolicity only results in a slight change in our final results [M. Zaluzny, Phys. Rev. B 43, 4511 (1991)].
[CrossRef]

1990 (1)

S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
[CrossRef] [PubMed]

1987 (1)

1976 (1)

O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. S. Lee and K. J. Malloy, IEEE J. Quantum Electron. QE-30, 85 (1994).
[CrossRef]

J. B. Khurgin and E. Rosencher, IEEE J. Quantum Electron. QE-32, 1882 (1996).
[CrossRef]

J. Appl. Phys. (1)

J. B. Khurgin, G. Sun, L. R. Friedman, and R. A. Soref, J. Appl. Phys. 78, 7398 (1995).
[CrossRef]

J. Nonlin. Opt. Phys. Mater. (1)

Y. Zhao, D. Huang, and C. Wu, J. Nonlin. Opt. Phys. Mater. 4, 261 (1995).
[CrossRef]

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

Opt. Lett. (2)

Phys. Rev. A (4)

D. Huang and Y. Zhao, Phys. Rev. A 51, 1617 (1995).
[CrossRef] [PubMed]

A. Liu and O. Keller, Phys. Rev. A 177, 441 (1993).

G. S. Agarwal, Phys. Rev. A 51, R2711 (1995).
[CrossRef]

A. Liu, Phys. Rev. A 56, 3206 (1997).
[CrossRef]

Phys. Rev. B (4)

Since the local-field (depolarization) effect is included in our formalism, it is expected that inclusion of the conduction band nonparabolicity only results in a slight change in our final results [M. Zaluzny, Phys. Rev. B 43, 4511 (1991)].
[CrossRef]

O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976).
[CrossRef]

A. Liu, Phys. Rev. B 50, 8569 (1994).
[CrossRef]

M. Zaluzny, Phys. Rev. B 47, 3995 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990).
[CrossRef] [PubMed]

Phys. Scr. (1)

A. Liu and O. Keller, Phys. Scr. 52, 116 (1995).
[CrossRef]

Other (3)

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).

G. Bastard, Wave Mechanics Applied to Semiconductor Heterostructures (Les Editions de Physique, Paris, 1988).

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific, Singapore, 1990).

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Figures (3)

Fig. 1
Fig. 1

Schematic diagram showing three closely spaced subbands (|1〉, |2〉, and |3〉) of a QW near-resonantly coupled by a strong pump field of frequency ωc and probed by a weak probe field of frequency ω.

Fig. 2
Fig. 2

Probe optical absorption (Ap) spectra of a QW system at the pump photon energy of ωc=124 meV for different pump field strengths, i.e., Ez0c=0 (curve 1), 8 (curve 2), 12 (curve 3), and 20 kV/cm (curve 4).

Fig. 3
Fig. 3

Probe optical absorption (Ap) spectra of a QW system at the pump photon energy of ωc=136 meV for different pump field strengths, i.e., Ez0c=0 (curve 1), 8 (curve 2), 12 (curve 3), and 20 kV/cm (curve 4).

Equations (86)

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i ρt=[H, ρ],
H=H0+H+Hrandom,
H=e2m* (pA+Ap).
A(r, t)=A(r, ω)exp(-iωt)+A*(r, ω)exp(iωt).
E(ω, r)=E(ω, q; z)exp(iqr).
|nk=12π exp(ikr)ψn(z),
H0|nk=En(k)|nk.
i ρnnt=-iτ1 [ρnn-ρnne]+m(Hnmρmn-ρnmHmn),
i ρnnt=[ωnn(k)-i/τ2]ρnn+m(Hnmρmn-ρnmHmn),
Hnm=-QWA(z)jmn(z)dz,
jmn(z)=-e2im* 2ikψn*(z)ψm(z)+ezψn*(z) dψm(z)dz-ψm(z) dψn*(z)dz
D21ρ21(ω)=Δρ12(0)H21(ω)+Δρ12(ω-ωc)×H21(ωc)+H23(-ωc)ρ31(ω+ωc),
D31ρ31(ω+ωc)=H32(ωc)ρ21(ω)-H21(ωc)ρ32(ω)+H32(ω)ρ21(ωc)-H21(ω)ρ32(ωc),
D32ρ32(ω)=Δρ23(0)H32(ω)+Δρ23(ω-ωc)×H32(ωc)-ρ31(ω+ωc)H12(-ωc),
AΔρ12(ω-ωc)=2H12(-ωc)ρ21(ω)-2H21(ω)×ρ12(-ωc)-2H21(ωc)×ρ12(ω-2ωc)+H32(ω)ρ23(-ωc)+H32(ωc)ρ23(ω-2ωc)-H23(-ωc)ρ32(ω),
AΔρ23(ω-ωc)=H21(ω)ρ12(-ωc)+H21(ωc)×ρ12(ω-2ωc)-H12(-ωc)×ρ21(ω)+2H23(-ωc)ρ32(ω)-2H32(ω)ρ23(-ωc)-2H32(ωc)ρ23(ω-2ωc),
B12ρ12(ω-2ωc)=-Δρ12(ω-ωc)H12(-ωc)-ρ13(-2ωc)H32(ω),
B23ρ23(ω-2ωc)=-Δρ23(ω-ωc)H23(-ωc)+ρ13(-2ωc)H21(ω).
Δρijρii-ρjj,
D21=[ω-ω21(k)+i/τ2],
D32=[ω-ω32(k)+i/τ2],
D31=[ω+ωc-ω31(k)+i/τ2],
A=(ω-ωc+i/τ1),
B12=[ω-2ωc-ω12+i/τ2],
B23=[ω-2ωc-ω23+i/τ2].
ρji(-ωc)=[ρij(ωc)]*,
Hji(-ωc)=[Hij(ωc)]*.
ρ21(ω)=A˜12H21(ω)+B˜12H32(ω),
ρ32(ω)=A˜23H32(ω)+B˜23H21(ω),
H21(ω)=e2m*ω N12,
H21(ωc)=e2m*ωc N12c,
H32(ω)=e2m*ω N23,
H32(ωc)=e2m*ωc N23c,
Nij=QWΦij(z)Ez(z)dz,i=1, 2; j=2, 3,
Nijc=QWΦij(z)Ezc(z)dz,i=1, 2; j=2, 3,
Φij(z)=ψi(z) dψj(z)dz-ψj(z) dψi(z)dz,
i=1, 2; j=2, 3.
Δ21ρ21(ωc)=Δρ12(0)H21(ωc)+H23(-ωc)ρ31(2ωc),
Δ32ρ32(ωc)=Δρ23(0)H32(ωc)-ρ31(2ωc)H12(-ωc),
Δ31ρ31(2ωc)=H32(ωc)ρ21(ωc)-ρ32(ωc)H21(ωc),
Δρ12(0)=Δρ12e+4τ1 Im[H12(-ωc)ρ21(ωc)]-2τ1 Im[H23(-ωc)ρ32(ωc)],
Δρ23(0)=Δρ23e+4τ1 Im[H23(-ωc)ρ32(ωc)]-2τ1 Im[H12(-ωc)ρ21(ωc)],
Δρijeρiie-ρjje,
Δ21=[ωc-ω21(k)+i/τ2],
Δ32=[ωc-ω32(k)+i/τ2],
Δ31=[2ωc-ω31(k)+i/τ2].
ρ21(ωc)=H21(ωc)Δρ12(0)-Δρ23(0)|H32(ωc)|2Δ32Δ31-|H21(ωc)|2×Δ31[Δ31Δ32-|H21(ωc)|2][Δ21Δ31-|H32(ωc)|2][Δ31Δ32-|H21(ωc)|2]-|H21(ωc)|2|H32(ωc)|2,
ρ32(ωc)=H32(ωc)Δρ23(0)-Δρ12(0)|H21(ωc)|2Δ31Δ21-|H32(ωc)|2×Δ31[Δ31Δ21-|H32(ωc)|2][Δ21Δ31-|H32(ωc)|2][Δ31Δ32-|H21(ωc)|2]-|H21(ωc)|2|H32(ωc)|2.
Δρ12(0)=1-4τ1 Im(X3)-2τ1 Im(X2)/D,
Δρ23(0)=-2τ1 [2 Im(X2)+Im(X1)]/D,
D=1+12τ122 [Im(X1)Im(X3)-Im(X2)2]-4τ1 [Im(X1)+Im(X2)+Im(X3)],
X1=Δ31[Δ32Δ31-|H21(ωc)|2]|H21(ωc)|2[Δ31Δ21-|H32(ωc)|2][Δ31Δ32-|H21(ωc)|2]-|H21(ωc)|2|H32(ωc)|2,
X2=Δ31|H21(ωc)|2|H32(ωc)|2[Δ31Δ21-|H32(ωc)|2][Δ31Δ32-|H21(ωc)|2]-|H21(ωc)|2|H32(ωc)|2,
X3=Δ31[Δ31Δ21-|H32(ωc)|2]|H32(ωc)|2[Δ31Δ21-|H32(ωc)|2][Δ31Δ32-|H21(ωc)|2]-|H21(ωc)|2|H32(ωc)|2.
Jzc(z)=Ns[ρ21(ωc)J12(z)+ρ32(ωc)J23(z)],
Jz(z)=Ns[ρ21(ω)J12(z)+ρ32(ω)J23(z)],
Jij(z)=ie2m* Φij(z),i=1, 2; j=2, 3,
Ezc(z)=Ez0c exp(iqcz)-iμ0ωcQWGzzc(z, z)Jzc(z)dz,
Gzzc(z, z)=c0ωc2 (qc)22iBqc exp(iqc|z-z|)+c0ωc2 1B δ(z-z)
Ez(z)=Ez0 exp(iqz)-iμ0ωQWGzz(z, z)Jz(z)dz.
N12c=Ez0c S12c(1-α23K23,23c)+S23cα23K12,23c(1-α12K12,12c)(1-α23K23,23c)-α12α23K12,23cK23,12c,
N23c=Ez0c S23c(1-α12K12,12c)+S12cα12K23,12c(1-α12K12,12c)(1-α23K23,23c)-α12α23K12,23cK23,12c,
α12=μ0e22Ns(2m*)2 Δρ12(0)-Δρ23(0)|H32(ωc)|2Δ32Δ31-|H21(ωc)|2×Δ31[Δ31Δ32-|H21(ωc)|2][Δ21Δ31-|H32(ωc)|2][Δ31Δ32-|H21(ωc)|2]-|H21(ωc)|2|H32(ωc)|2,
α23=μ0e22Ns(2m*)2 Δρ23(0)-Δρ12(0)|H21(ωc)|2Δ31Δ21-|H32(ωc)|2×Δ31[Δ31Δ21-|H32(ωc)|2](Δ21Δ31-|H32(ωc)|2)(Δ31Δ32-|H21(ωc)|2)-|H21(ωc)|2|H32(ωc)|2,
Kij,lmc=QWΦij(z)Gzzc(z, z)Φlm(z)dzdz,
Sijc=QW exp(iqcz)Φij(z)dz.
N12N˜12Ez0=Ez0F [S12(1-K23,12B12-K23,23A23)+S23(K12,12B12+K12,23A23)],
N23N˜23Ez0=Ez0F [S23(1-K12,12A12-K12,23B23)+S12(K23,12A12+K23,23B23)],
F=(1-K12,12A12-K12,23B23)(1-K23,12B12-K23,23A23)-(K12,12B12+K12,23A23)×(K23,12A12+K23,23B23),
A12=μ0e22Ns(2m*)2 A˜12,
B12=μ0e22Ns(2m*)2 B˜12,
A23=μ0e22Ns(2m*)2 A˜23,
B23=μ0e22Ns(2m*)2 B˜23.
rp=(A12N˜12+B12N˜23)R12+(A23N˜23+B23N˜12)R23,
tp=1+(A12N˜12+B12N˜23)T12+(A23N˜23+B23N˜12)T23,
Rij=c0ω2 q22iBq QW exp(iqz)Φij(z)dz,
i=1, 2; j=2, 3,
Tij=c0ω2 q22iBq QW exp(-iqz)Φij(z)dz,
i=1, 2; j=2, 3.
Ap=1-|rp|2-|tp|2.
-22m* d2ψndz2+V(z)ψn(z)=Enψn(z),
d2VH(z)dz2=-e20r [n(z)-ND(z)],
VXC(z)=-e24π20raB* 9π41/3×1rs(z)+0.0545 ln1+11.4rs(z),
aB*=4π0r2m*e2,
rs(z)=4π3 (aB*)3n(z)-1/3.
n(z)=Ns|ψ1(z)|2.

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