Abstract

We propose a self-consistent many-body theory for coupling the ultrafast dipole-transition and carrier-plasma dynamics in a linear array of quantum wires with the scattering and absorption of ultrashort laser pulses. The quantum-wire non-thermal carrier occupations are further driven by an applied DC electric field along the wires in the presence of resistive forces from intrinsic phonon and Coulomb scattering of photo-excited carriers. The same strong DC field greatly modifies the non-equilibrium properties of the induced electron-hole plasma coupled to the propagating light pulse, while the induced longitudinal polarization fields of each wire significantly alters the nonlocal optical response from neighboring wires. Here, we clarify several fundamental physics issues in this laser-coupled quantum wire system, including laser pulse influence on local transient photo-currents, photoluminescence spectra, and the effect of nonlinear transport in a micro-scale system on laser pulse propagation. Meanwhile, we also anticipate some applications from this work, such as specifying the best combination of pulse sequence through a quantum-wire array to generate a desired THz spectrum and applying ultra-fast optical modulations to nonlinear carrier transport in nanowires.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  7. R. Buschlingern, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
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    [Crossref]
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    [Crossref]
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  23. D. Huang and S. K. Lyo, “Photoluminescence spectra of n-doped double quantum wells in a parallel magnetic field,” Phys. Rev. B 59, 7600–7609 (1999).
    [Crossref]
  24. D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
    [Crossref]
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    [Crossref]
  26. D. Huang and D. A. Cardimona, “Effects of off-diagonal radiative-decay coupling on electron transitions in resonant double quantum wells,” Phys. Rev. A 64, 013822 (2001).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  32. D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. i. intrasubband excitation,” Phys. Rev. B 38, 13061–13068 (1988).
    [Crossref]
  33. D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. ii. intersubband excitation and effects of magnetic field and electron-phonon coupling,” Phys. Rev. B 38, 13069–13078 (1988).
    [Crossref]
  34. D. Huang and S. Zhou, “Intrasubband and intersubband collective excitations in a quasi-0+1-dimensional superlattice,” J. Physics: Condens. Matter 2, 501–504 (1990).
  35. M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
    [Crossref]
  36. A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
    [Crossref]

2017 (2)

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
[Crossref]

2016 (1)

2015 (2)

R. Buschlingern, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

2014 (1)

2010 (1)

2009 (2)

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706 (2009).
[Crossref]

D. Huang and G. Gumbs, “Coupled force-balance and scattering equations for nonlinear transport in quantum wires,” Phys. Rev. B 80, 033411 (2009).
[Crossref]

2008 (1)

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

2006 (2)

D. Huang, C. Rhodes, P. M. Alsing, and D. A. Cardimona, “Effects of longitudinal field on transmitted near field in doped semi-infinite semiconductors with a surface conducting sheet,” J. Appl. Phys. 100, 113711 (2006).
[Crossref]

R. Agarwal and C. Lieber, “Semiconductor nanowires: optics and optoelectronics,” Appl. Phys. A 85, 209 (2006).
[Crossref]

2005 (2)

D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
[Crossref]

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

2004 (1)

D. H. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, “High-field transport of electrons and radiative effects using coupled force-balance and fokker-planck equations beyond the relaxation-time approximation,” Phys. Rev. B 69, 075214 (2004).
[Crossref]

2002 (1)

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[Crossref]

2001 (1)

D. Huang and D. A. Cardimona, “Effects of off-diagonal radiative-decay coupling on electron transitions in resonant double quantum wells,” Phys. Rev. A 64, 013822 (2001).
[Crossref]

1999 (1)

D. Huang and S. K. Lyo, “Photoluminescence spectra of n-doped double quantum wells in a parallel magnetic field,” Phys. Rev. B 59, 7600–7609 (1999).
[Crossref]

1996 (2)

D. Huang and M. O. Manasreh, “Intersubband transitions in strained in0.07ga0.93as/al0.40ga0.60 as multiple quantum wells and their application to a two-colors photodetector,” Phys. Rev. B 54, 5620–5628 (1996).
[Crossref]

D. Huang and M. O. Manasreh, “Effects of the screened exchange interaction on the tunneling and landau gaps in double quantum wells,” Phys. Rev. B 54, 2044–2048 (1996).
[Crossref]

1992 (1)

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

1991 (1)

J. R. Kuklinski and S. Mukamel, “Generalized semiconductor bloch equations: Local fields and transient gratings,” Phys. Rev. B 44, 11253–11259 (1991).
[Crossref]

1990 (1)

D. Huang and S. Zhou, “Intrasubband and intersubband collective excitations in a quasi-0+1-dimensional superlattice,” J. Physics: Condens. Matter 2, 501–504 (1990).

1988 (3)

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. i. intrasubband excitation,” Phys. Rev. B 38, 13061–13068 (1988).
[Crossref]

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. ii. intersubband excitation and effects of magnetic field and electron-phonon coupling,” Phys. Rev. B 38, 13069–13078 (1988).
[Crossref]

M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342 (1988).
[Crossref]

1971 (1)

R. E. Fern and A. Onton, “Refractive index of alas,” J. Appl. Phys. 42, 3499–3500 (1971).
[Crossref]

Afzelius, M.

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

Agarwal, R.

R. Agarwal and C. Lieber, “Semiconductor nanowires: optics and optoelectronics,” Appl. Phys. A 85, 209 (2006).
[Crossref]

Alsing, P. M.

D. Huang, C. Rhodes, P. M. Alsing, and D. A. Cardimona, “Effects of longitudinal field on transmitted near field in doped semi-infinite semiconductors with a surface conducting sheet,” J. Appl. Phys. 100, 113711 (2006).
[Crossref]

D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
[Crossref]

D. H. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, “High-field transport of electrons and radiative effects using coupled force-balance and fokker-planck equations beyond the relaxation-time approximation,” Phys. Rev. B 69, 075214 (2004).
[Crossref]

Apostolova, T.

D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
[Crossref]

D. H. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, “High-field transport of electrons and radiative effects using coupled force-balance and fokker-planck equations beyond the relaxation-time approximation,” Phys. Rev. B 69, 075214 (2004).
[Crossref]

Appenzeller, J.

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

Bermel, P.

Binder, R.

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

Bjork, M. T.

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

Buschlingern, R.

R. Buschlingern, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Cardimona, D. A.

D. Huang, C. Rhodes, P. M. Alsing, and D. A. Cardimona, “Effects of longitudinal field on transmitted near field in doped semi-infinite semiconductors with a surface conducting sheet,” J. Appl. Phys. 100, 113711 (2006).
[Crossref]

D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
[Crossref]

D. H. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, “High-field transport of electrons and radiative effects using coupled force-balance and fokker-planck equations beyond the relaxation-time approximation,” Phys. Rev. B 69, 075214 (2004).
[Crossref]

D. Huang and D. A. Cardimona, “Effects of off-diagonal radiative-decay coupling on electron transitions in resonant double quantum wells,” Phys. Rev. A 64, 013822 (2001).
[Crossref]

Chatterjee, S.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

de Riedmatten, H.

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

Diels, J.-C.

J.-C. Diels and W. Rudolf, Ultrashort Laser Pulse Phenomenon: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale (Academic Press, 2006), 2nd ed.

Ell, C.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Ferguson, B.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[Crossref]

Fern, R. E.

R. E. Fern and A. Onton, “Refractive index of alas,” J. Appl. Phys. 42, 3499–3500 (1971).
[Crossref]

Forstmann, F.

F. Forstmann and R. R. Gerhardts, Metal Optics Near the Plasma Frequency (Springer-Verlag, 1986).
[Crossref]

Gerhardts, R. R.

F. Forstmann and R. R. Gerhardts, Metal Optics Near the Plasma Frequency (Springer-Verlag, 1986).
[Crossref]

Gibbs, H. M.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Gisin, N.

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

Gumbs, G.

A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
[Crossref]

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

D. Huang and G. Gumbs, “Coupled force-balance and scattering equations for nonlinear transport in quantum wires,” Phys. Rev. B 80, 033411 (2009).
[Crossref]

G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (John Wiley & Sons, 2011).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Inc., 2000), 3rd ed.

Haug, H.

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific Publishing Co. Pte. Ltd., 2009), 5th ed.
[Crossref]

Henneberger, K.

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

Herzel, F.

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

Hoyer, W.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Huang, D.

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
[Crossref]

D. Huang and G. Gumbs, “Coupled force-balance and scattering equations for nonlinear transport in quantum wires,” Phys. Rev. B 80, 033411 (2009).
[Crossref]

D. Huang, C. Rhodes, P. M. Alsing, and D. A. Cardimona, “Effects of longitudinal field on transmitted near field in doped semi-infinite semiconductors with a surface conducting sheet,” J. Appl. Phys. 100, 113711 (2006).
[Crossref]

D. Huang and D. A. Cardimona, “Effects of off-diagonal radiative-decay coupling on electron transitions in resonant double quantum wells,” Phys. Rev. A 64, 013822 (2001).
[Crossref]

D. Huang and S. K. Lyo, “Photoluminescence spectra of n-doped double quantum wells in a parallel magnetic field,” Phys. Rev. B 59, 7600–7609 (1999).
[Crossref]

D. Huang and M. O. Manasreh, “Intersubband transitions in strained in0.07ga0.93as/al0.40ga0.60 as multiple quantum wells and their application to a two-colors photodetector,” Phys. Rev. B 54, 5620–5628 (1996).
[Crossref]

D. Huang and M. O. Manasreh, “Effects of the screened exchange interaction on the tunneling and landau gaps in double quantum wells,” Phys. Rev. B 54, 2044–2048 (1996).
[Crossref]

D. Huang and S. Zhou, “Intrasubband and intersubband collective excitations in a quasi-0+1-dimensional superlattice,” J. Physics: Condens. Matter 2, 501–504 (1990).

Huang, D. H.

D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
[Crossref]

D. H. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, “High-field transport of electrons and radiative effects using coupled force-balance and fokker-planck equations beyond the relaxation-time approximation,” Phys. Rev. B 69, 075214 (2004).
[Crossref]

G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (John Wiley & Sons, 2011).
[Crossref]

Huang, D.-h.

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. ii. intersubband excitation and effects of magnetic field and electron-phonon coupling,” Phys. Rev. B 38, 13069–13078 (1988).
[Crossref]

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. i. intrasubband excitation,” Phys. Rev. B 38, 13061–13068 (1988).
[Crossref]

Iurov, A.

A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
[Crossref]

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1975).

Jagadish, C.

Khan, M. R.

Khitrova, G.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Kira, M.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Knoch, J.

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

Koch, S. W.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342 (1988).
[Crossref]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific Publishing Co. Pte. Ltd., 2009), 5th ed.
[Crossref]

Kuklinski, J. R.

J. R. Kuklinski and S. Mukamel, “Generalized semiconductor bloch equations: Local fields and transient gratings,” Phys. Rev. B 44, 11253–11259 (1991).
[Crossref]

Kupec, J.

Lieber, C.

R. Agarwal and C. Lieber, “Semiconductor nanowires: optics and optoelectronics,” Appl. Phys. A 85, 209 (2006).
[Crossref]

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M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342 (1988).
[Crossref]

Lorke, M.

R. Buschlingern, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Lundstrom, M.

Lvovsky, A. I.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706 (2009).
[Crossref]

Lyo, S. K.

D. Huang and S. K. Lyo, “Photoluminescence spectra of n-doped double quantum wells in a parallel magnetic field,” Phys. Rev. B 59, 7600–7609 (1999).
[Crossref]

Manasreh, M. O.

D. Huang and M. O. Manasreh, “Effects of the screened exchange interaction on the tunneling and landau gaps in double quantum wells,” Phys. Rev. B 54, 2044–2048 (1996).
[Crossref]

D. Huang and M. O. Manasreh, “Intersubband transitions in strained in0.07ga0.93as/al0.40ga0.60 as multiple quantum wells and their application to a two-colors photodetector,” Phys. Rev. B 54, 5620–5628 (1996).
[Crossref]

Maradudin, A. A.

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

Mokkapati, S.

Mosor, S.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Mukamel, S.

J. R. Kuklinski and S. Mukamel, “Generalized semiconductor bloch equations: Local fields and transient gratings,” Phys. Rev. B 44, 11253–11259 (1991).
[Crossref]

Onton, A.

R. E. Fern and A. Onton, “Refractive index of alas,” J. Appl. Phys. 42, 3499–3500 (1971).
[Crossref]

Pan, W.

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

Paul, A. E.

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

Peschel, U.

R. Buschlingern, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

Rhodes, C.

D. Huang, C. Rhodes, P. M. Alsing, and D. A. Cardimona, “Effects of longitudinal field on transmitted near field in doped semi-infinite semiconductors with a surface conducting sheet,” J. Appl. Phys. 100, 113711 (2006).
[Crossref]

Riel, H.

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

Riess, W.

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

Rudolf, W.

J.-C. Diels and W. Rudolf, Ultrashort Laser Pulse Phenomenon: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale (Academic Press, 2006), 2nd ed.

Sanders, B. C.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706 (2009).
[Crossref]

Schmid, H.

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

Scott, D.

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

Shen, Y. R.

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, 1984).

Stolz, H.

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

Stoop, R. L.

Taflove, A.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Inc., 2000), 3rd ed.

Tittel, W.

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706 (2009).
[Crossref]

Wang, X.

Witzigmann, B.

Xu, J.

X.-C. Zhang and J. Xu, Introduction to THz Waves Photonics (Springer, 2010).
[Crossref]

Zhang, X.-C.

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[Crossref]

X.-C. Zhang and J. Xu, Introduction to THz Waves Photonics (Springer, 2010).
[Crossref]

Zhemchuzhna, L.

A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
[Crossref]

Zhou, S.

D. Huang and S. Zhou, “Intrasubband and intersubband collective excitations in a quasi-0+1-dimensional superlattice,” J. Physics: Condens. Matter 2, 501–504 (1990).

Zhou, S.-x.

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. ii. intersubband excitation and effects of magnetic field and electron-phonon coupling,” Phys. Rev. B 38, 13069–13078 (1988).
[Crossref]

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. i. intrasubband excitation,” Phys. Rev. B 38, 13061–13068 (1988).
[Crossref]

Appl. Phys. A (1)

R. Agarwal and C. Lieber, “Semiconductor nanowires: optics and optoelectronics,” Appl. Phys. A 85, 209 (2006).
[Crossref]

IEEE Trans. Electron Devices (1)

J. Appenzeller, J. Knoch, M. T. Bjork, H. Riel, H. Schmid, and W. Riess, “Toward nanowire electronics,” IEEE Trans. Electron Devices 55, 2827–2845 (2008).
[Crossref]

J. Appl. Phys. (3)

D. Huang, C. Rhodes, P. M. Alsing, and D. A. Cardimona, “Effects of longitudinal field on transmitted near field in doped semi-infinite semiconductors with a surface conducting sheet,” J. Appl. Phys. 100, 113711 (2006).
[Crossref]

R. E. Fern and A. Onton, “Refractive index of alas,” J. Appl. Phys. 42, 3499–3500 (1971).
[Crossref]

A. Iurov, L. Zhemchuzhna, G. Gumbs, and D. Huang, “Exploring interacting floquet states in black phosphorus: Anisotropy and bandgap laser tuning,” J. Appl. Phys. 122, 124301 (2017).
[Crossref]

J. Physics: Condens. Matter (1)

D. Huang and S. Zhou, “Intrasubband and intersubband collective excitations in a quasi-0+1-dimensional superlattice,” J. Physics: Condens. Matter 2, 501–504 (1990).

Nat. Mater. (1)

B. Ferguson and X.-C. Zhang, “Materials for terahertz science and technology,” Nat. Mater. 1, 26–33 (2002).
[Crossref]

Nat. Photon. (1)

A. I. Lvovsky, B. C. Sanders, and W. Tittel, “Optical quantum memory,” Nat. Photon. 3, 706 (2009).
[Crossref]

Opt. Express (3)

Phys. Rev. A (2)

K. Henneberger, F. Herzel, S. W. Koch, R. Binder, A. E. Paul, and D. Scott, “Spectral hole burning and gain saturation in short-cavity semiconductor lasers,” Phys. Rev. A 45, 1853–1859 (1992).
[Crossref] [PubMed]

D. Huang and D. A. Cardimona, “Effects of off-diagonal radiative-decay coupling on electron transitions in resonant double quantum wells,” Phys. Rev. A 64, 013822 (2001).
[Crossref]

Phys. Rev. B (13)

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. i. intrasubband excitation,” Phys. Rev. B 38, 13061–13068 (1988).
[Crossref]

D.-h. Huang and S.-x. Zhou, “Theoretical investigation of collective excitations in hgte/cdte superlattices. ii. intersubband excitation and effects of magnetic field and electron-phonon coupling,” Phys. Rev. B 38, 13069–13078 (1988).
[Crossref]

D. H. Huang, T. Apostolova, P. M. Alsing, and D. A. Cardimona, “High-field transport of electrons and radiative effects using coupled force-balance and fokker-planck equations beyond the relaxation-time approximation,” Phys. Rev. B 69, 075214 (2004).
[Crossref]

D. Huang and G. Gumbs, “Coupled force-balance and scattering equations for nonlinear transport in quantum wires,” Phys. Rev. B 80, 033411 (2009).
[Crossref]

D. Huang and S. K. Lyo, “Photoluminescence spectra of n-doped double quantum wells in a parallel magnetic field,” Phys. Rev. B 59, 7600–7609 (1999).
[Crossref]

D. H. Huang, P. M. Alsing, T. Apostolova, and D. A. Cardimona, “Coupled energy-drift and force-balance equations for high-field hot-carrier transport,” Phys. Rev. B 71, 195205 (2005).
[Crossref]

W. Hoyer, C. Ell, M. Kira, S. W. Koch, S. Chatterjee, S. Mosor, G. Khitrova, H. M. Gibbs, and H. Stolz, “Many-body dynamics and exciton formation studied by time-resolved photoluminescence,” Phys. Rev. B 72, 075324 (2005).
[Crossref]

M. Lindberg and S. W. Koch, “Effective bloch equations for semiconductors,” Phys. Rev. B 38, 3342 (1988).
[Crossref]

R. Buschlingern, M. Lorke, and U. Peschel, “Light-matter interaction and lasing in semiconductor nanowires: A combined finite-difference time-domain and semiconductor bloch equation approach,” Phys. Rev. B 91, 045203 (2015).
[Crossref]

A. Iurov, D. Huang, G. Gumbs, W. Pan, and A. A. Maradudin, “Effects of optical polarization on hybridization of radiative and evanescent field modes,” Phys. Rev. B 96, 081408 (2017).
[Crossref]

J. R. Kuklinski and S. Mukamel, “Generalized semiconductor bloch equations: Local fields and transient gratings,” Phys. Rev. B 44, 11253–11259 (1991).
[Crossref]

D. Huang and M. O. Manasreh, “Intersubband transitions in strained in0.07ga0.93as/al0.40ga0.60 as multiple quantum wells and their application to a two-colors photodetector,” Phys. Rev. B 54, 5620–5628 (1996).
[Crossref]

D. Huang and M. O. Manasreh, “Effects of the screened exchange interaction on the tunneling and landau gaps in double quantum wells,” Phys. Rev. B 54, 2044–2048 (1996).
[Crossref]

Phys. Today (1)

M. Afzelius, N. Gisin, and H. de Riedmatten, “Quantum memory for photons,” Phys. Today 68, 42–47 (2015).
[Crossref]

Other (9)

X.-C. Zhang and J. Xu, Introduction to THz Waves Photonics (Springer, 2010).
[Crossref]

J.-C. Diels and W. Rudolf, Ultrashort Laser Pulse Phenomenon: Fundamentals, Techniques, and Applications on a Femtosecond Time Scale (Academic Press, 2006), 2nd ed.

K. T. Tsen, ed., Ultrafast Physical Processes in Semiconductors, vol. 67 (Academic Press, 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA, 2001).
[Crossref]

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors (World Scientific Publishing Co. Pte. Ltd., 2009), 5th ed.
[Crossref]

G. Gumbs and D. H. Huang, Properties of Interacting Low-Dimensional Systems (John Wiley & Sons, 2011).
[Crossref]

Y. R. Shen, The Principles of Nonlinear Optics (John Wiley & Sons, 1984).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Inc., 2000), 3rd ed.

F. Forstmann and R. R. Gerhardts, Metal Optics Near the Plasma Frequency (Springer-Verlag, 1986).
[Crossref]

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, 1975).

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Figures (14)

Fig. 1
Fig. 1 Schematic of a model system which consists of biased quantum wires extending along the y direction and displayed in the x direction by a linear array. An incident laser pulse, with a Gaussian spatial profile in the x direction and its electric and magnetic fields along x and z directions, propagates along the y direction and generates e–h pairs in quantum wires by interacting with them. Additionally, induced electrons and holes in quantum wires are driven by DC electric fields.
Fig. 2
Fig. 2 Calculated n j , k e , h ( t ) [in (a)] and n j , k h ( t ) [in (b)] from Eqs. (7a) and (7b) as functions of carrier wave number k for electrons (e) and holes (h) within the central quantum wire at different times t. Here, results for electrons and holes are shown for different moments at t = 120, 140, 160, 335, 670 fs and t = 1, 1.6, 10 ps for a 40 fs light pulse interacting significantly with electrons in quantum wires within the time interval t ∈ 100 − 180 fs.
Fig. 3
Fig. 3 Calculated Re [ p j , k , k σ ( t ) ] (solid) and Im [ p j , k , k σ ( t ) ] (dashed) from Eq. (7c) as functions of carrier wave number k for optical coherence of electron-hole pairs within the central quantum wire at different moments at t = 120, 140, 160, 180 fs for a 40 fs light pulse interacting significantly with electrons in quantum wires within the time interval t ∈ 100 − 180 fs. Here, results for σ = x, y, corresponding to two directions perpendicular and parallel to quantum wires, are shown in (a) with k′ = k and (b) with k′ = q0, respectively.
Fig. 4
Fig. 4 Calculated diagonal dephasing rates for electrons Δ j , k e ( t ) (solid) and holes Δ j , k h ( t ) (dashed) from Eqs. (47) and (48) are presented as functions of wave number k for electrons and holes within the central quantum wire, where results for electrons and holes are shown for different moments at t = 120, 140, 160, 180 fs for a 40 fs light pulse interacting with quantum wires within the time interval t ∈ 100 − 180 fs.
Fig. 5
Fig. 5 Calculated photo-generated carrier density n j , 1 D ( t ) = n j , 1 D e ( t ) = n j , 1 D h ( t ) = N j e , h ( t ) / [see its expression right after Eqs. (23a) and (23b)] as a function of t for electrons (e) and holes (h) within the central quantum wire for the bandgap εG = 1.50 eV (black) and εG = 1.54 eV (red). Here, a 40 fs light pulse interacts significantly with quantum wires within the time interval t ∈ 100 − 180 fs.
Fig. 6
Fig. 6 Calculated time-evolution of photoluminescence spectra ��j,pl0 | t) (logarithmic scale) from Eq. (20) for spontaneous emission within the central quantum wire at different times t = 120, 140, 180, 667 fs and t = 1.6 ps. Here, ħΩ0 is the energy of spontaneously emitted photons and a 40 fs light pulse interacts significantly with quantum wires within the time interval t ∈ 100 − 180 fs.
Fig. 7
Fig. 7 Calculated effective temperatures for electrons Tj,e(t) (red) and holes Tj,h(t) (blue) [see their expressions right after Eq. (19)] as functions of t within the central quantum wire for two values of DC electric field Edc = 1 (solid) and 3 kV/cm (dashed). Both the expanded (a) and the complete (b) views are presented. Here, a 40 fs light pulse interacts significantly with quantum wires within the time interval t ∈ 100 − 180 fs.
Fig. 8
Fig. 8 Calculated mobilities for electrons | μ j e ( t ) | [(a),(c)] and holes μ j h ( t ) [(b),(d)] from Eq. (18) as functions of t within the central quantum wire with six values of DC electric field Edc from 0.5 to 3.0 kV/cm in steps of 0.5 kV/cm. Both results for a 40 fs [(a),(b)] and a 100 fs [(c),(d)] pulse are presented. Here, the light pulse interacts significantly with quantum wires within the time interval t ∈ 100 – 180 fs.
Fig. 9
Fig. 9 Calculated photo-current Ij,ph(t) = Jj,ph(t)(2δ0/α) from Eq. (17) at Edc = 1 kV/cm as a function of t within the central quantum wire. The expanded (a) and the complete (b) views are presented. Here, a 40 fs light pulse interacts significantly with quantum wires within the time interval t ∈ 100 – 180 fs.
Fig. 10
Fig. 10 Calculated real and imaginary parts of P ˜ j x ( q , t ) [(a) (c),(d)] and P ˜ j y ( q , t ) [(b),(e),(f)] from Eq. (21b) as a function of wave number q [(a), (b)] (real-solid, imaginary-dashed) at different times t = 120, 140, 160 fs within the central quantum wire as well as a function of time t [(c)–(f)] (real-black, imaginary-red) at q/q0 = 1 [(c),(e)] and 10 [(d),(f)]. Here, the light pulse interacts significantly with quantum wires within the time interval t ∈ 100−180 fs.
Fig. 11
Fig. 11 (Density plots for spatial distributions of both transverse [(a),(b)] and longitudinal [(c),(d)] local polarization fields at the moment of t = 140 fs that a 40 fs pulse peak is passing through the middle of a quantum-wire array (y = 0), where transverse- and longitudinal-polarization-field components along both the x [(a),(c)] and y [(b),(d)] directions are displayed. The linear array consists of three quantum wires displaced in the x direction at x = 0 and ±125 nm, respectively.
Fig. 12
Fig. 12 Density plots for spatial distributions of both transverse [(a),(b)] and longitudinal [(c),(d)] propagating electric fields at the moment of t = 140 fs that a 40 fs pulse peak is passing through the middle of a quantum-wire array (y = 0), where transverse- and longitudinal-field components along both the x [(a),(c)] and y [(b),(d)] directions are presented. The array used is the same as that in Fig. 11.
Fig. 13
Fig. 13 Density plots for spatial distributions of the longitudinal electric fields at the moment of t = 140 fs that a 40 fs pulse peak is passing through the middle of a quantum-wire array, where longitudinal-field components along both the x [(a)] and y [(b)] directions are presented. The linear array consists of ten quantum wires displaced in the x direction at intervals of 40 nm.
Fig. 14
Fig. 14 Calculated intensity ratios for incident (black), reflected (red) and transmitted (blue) in both logarithm [(a),(c)] and linear [(b),(d)] scales for 40 fs light pulses with peak intensities 6.2 GW/cm2 [(a),(b)] and 0.62 kW/cm2 [(c),(d)] from Eqs. (27a) and (27b) as functions of Fourier frequency Ω.

Tables (2)

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Table 1 Parameters for AlAs host semiconductor

Tables Icon

Table 2 Parameters for GaAs quantum wires

Equations (80)

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i q D ˜ ( q , t ) = ρ ˜ qw ( q , t ) ,
i q B ˜ ( q , t ) = 0 ,
i q × E ˜ ( q , t ) = t B ˜ ( q , t ) ,
i q × H ˜ ( q , t ) = t D ˜ ( q , t ) .
D ˜ ( q , t ) = e ^ q [ ρ ˜ qw ( q , t ) i q ] ,
H ˜ ( q , t ) = 0 ,
D ˜ ( q , t ) t = i q × H ˜ ( q , t ) ,
H ˜ ( q , t ) t = i 0 c 2 q × E ˜ ( q , t ) ,
E ˜ , ( q , t ) = D ˜ , ( q , t ) i P ˜ i , ( q , t ) + P ˜ qw , ( q , t ) 0 b .
ρ ˜ qw ( q , t ) = j ρ ˜ j 1 D ( q , t ) e i q R j q 2 / 4 α 2 ,
P ˜ qw { , } ( q , t ) = σ = x , y 𝒫 ˜ qw σ ( q , t ) 𝒢 ˜ { , } σ ( q ) = j e i q R j q 2 / 4 α 2 σ = x , y P ˜ j σ ( q , t ) 𝒢 ˜ { , } σ ( q ) ,
𝒢 ˜ x ( q ) = ( e ^ q e ^ x ) e ^ q = q q 2 + q 2 ( q e ^ x + q e ^ y ) ,
𝒢 ˜ y ( q ) = ( e ^ q e ^ y ) e ^ q = q q 2 + q 2 ( q e ^ x + q e ^ y ) ,
𝒢 ˜ x ( q ) = ( e ^ q × e ^ q × e ^ x ) = q q 2 + q 2 ( q e ^ x q e ^ y ) ,
𝒢 ˜ y ( q ) = ( e ^ q × e ^ q × e ^ y ) = q q 2 + q 2 ( q e ^ x q e ^ y ) ,
d n j , k e ( t ) d t = 2 k Im { p j , k , k ( t ) Ω j , k , k ( t ) } + n j , k e ( t ) t | rel ,
d n j , k h ( t ) d t = 2 k Im { p j , k , k ( t ) Ω j , k , k ( t ) } + n j , k h ( t ) t | rel ,
i d p j , k , k ( t ) d t = [ ε k e + ε k h + ε G + Δ ε j , k e + + Δ ε j , k h i Δ j , k , k eh ( t ) ] p j , k , k ( t ) [ 1 n k e ( t ) n k h ( t ) ] Ω j , k , k ( t ) + i q 0 Λ j , k , q e ( t ) p j , k + q , k ( t ) + i q 0 Λ j , k , q h ( t ) p j , k , k + q ( t ) ,
Δ ε j , k e = 2 q n j , q e ( t ) V k , q ; q , k ee q k n j , q e ( t ) V k , q ; k , q ee 2 q n j , q h ( t ) V k , q ; q , k eh ,
Δ ε j , k h = 2 q n j , q h ( t ) V k , q ; q , k hh q k n j , q h ( t ) V k , q ; k , q hh 2 q n j , q e ( t ) V q , k ; k , q eh ,
1 D ( q , t ) = 1 lim ω 0 2 β m e * π 2 q ln { ω 2 [ Ω e ( q , t ) ] 2 ω 2 [ Ω e + ( q , t ) ] 2 } K 0 ( q R e ) lim ω 0 2 β m h * π 2 q ln { ω 2 [ Ω h ( q , t ) ] 2 ω 2 [ Ω h + ( q , t ) ] 2 } K 0 ( q R h ) ,
n j , k e ( t ) t | rel = n j , k e ( t ) t | scat j , sp ( k , t ) n j , k e ( t ) n j , k h ( t ) + j e ( t ) n j , k e ( t ) k ,
n j , k h ( t ) t | rel = n j , k h ( t ) t | scat j , sp ( k , t ) n j , k e ( t ) n j , k h ( t ) + j h ( t ) n j , k h ( t ) k .
n j , k e ( t ) t | scat = W j , k e , ( in ) ( t ) [ 1 n j , k e ( t ) ] W j , k e , ( out ) ( t ) n j , k e ( t ) ,
n j , k h ( t ) t | scat = W j , k h , ( in ) ( t ) [ 1 n j , k h ( t ) ] W j , k h , ( out ) ( t ) n j , k h ( t ) ,
j , sp ( k , t ) = 3 d cv 2 0 r 0 d ω { ω ρ 0 ( ω ) L ( ω ε G ε k e ε k h ε j , c ( k , t ) , γ eh ) × M ( ω ε G ε j , c ( k = 0 , t ) , γ eh ) } ,
ε j , c ( k , t ) = q n j , q e ( t ) ( V k , q ; q , k ee V k , q ; k , q ee ) + q n j , q h ( t ) ( V k , q ; q , k hh V k , q ; k , q hh ) q k n j , q e ( t ) V q , k ; k , q eh q k n j , q h ( t ) V k , q ; q , k eh V k , k ; k , k eh ,
j e ( t ) = e E dc 2 k , q q { Θ j , e em ( k , q , t ) Θ j , e abs ( k , q , t ) } ,
j h ( t ) = + e E dc 2 k , q q { Θ j , h em ( k , q , t ) Θ j , h abs ( k , q , t ) } ,
Θ j , e em ( k , q , t ) = 4 π | V k , k q ep | 2 n j , k e ( t ) [ 1 n j , k q e ( t ) ] [ N 0 ( Ω ph ) + 1 ] × L ( ε k q e ε k e + Ω ph q v j e ( t ) , γ e ) θ ( Ω ph q v j e ( t ) ) ,
Θ j , e abs ( k , q , t ) = 4 π | V k , k q ep | 2 n j , k q e ( t ) [ 1 n j , k e ( t ) ] N 0 ( Ω ph ) × L ( ε k e ε k q e Ω ph + q v j e ( t ) , γ e ) θ ( Ω ph q v j e ( t ) ) ,
Θ j , h em ( k , q , t ) = 4 π | V k , k q hp | 2 n j , k h ( t ) [ 1 n j , k q h ( t ) ] [ N 0 ( Ω ph ) + 1 ] × L ( ε k q h ε k h + Ω ph q v j h ( t ) , γ h ) θ ( Ω ph q v j h ( t ) ) ,
Θ j , h abs ( k , q , t ) = 4 π | V k , k q hp | 2 n j , k q h ( t ) [ 1 n j , k h ( t ) ] N 0 ( Ω ph ) × L ( ε k h ε k q h Ω ph + q v j h ( t ) , γ h ) θ ( Ω ph q v j h ( t ) ) ,
J j , ph ( t ) = e α 2 δ 0 [ n j , 1 D h ( t ) v j h ( t ) n j , 1 D e ( t ) v j e ( t ) ] e ^ w [ σ j h ( t ) + σ j e ( t ) ] E dc e ^ w ,
v j e ( h ) ( t ) = 2 N j e , h ( t ) k d E ¯ j , k e ( h ) ( t ) d k n j , k e ( h ) ( t ) μ j e ( h ) ( t ) E dc ,
𝒬 j tot ( t ) = 𝒬 j e ( t ) + 𝒬 j h ( t ) 2 k E ¯ j , k e ( t ) n j e ( t ) + 2 k E ¯ j , k h ( t ) n j h ( t ) ,
𝒫 j , pl ( Ω 0 | t ) = 3 d cv 2 0 r Ω 0 ρ 0 ( Ω 0 ) k n j , k e ( t ) n j , k h ( t ) L ( Ω 0 ε G ε k e ε k h ε j , c ( k , t ) , γ eh ) × M ( Ω 0 ε G ε j , c ( k = 0 , t ) , γ eh ) ,
P ˜ qw ( q , t ) = σ = x , y P ˜ qw σ ( q , t ) e ^ d σ = j e i q R j q 2 / 4 α 2 σ = x , y P ˜ j σ ( q , t ) e ^ d σ ,
P ˜ j σ ( q , t ) = d cv α 2 δ 0 k p j , k + q , k σ ( t ) + H . C . ,
ρ ˜ j 1 D ( q , t ) = ρ ˜ j h ( q , t ) + ρ ˜ j e ( q , t ) ,
ρ ˜ j h ( q , t ) = e α N j e ( t ) δ 0 k , k σ = x , y p j , k , k q σ ( t ) [ p j , k , k σ ( t ) ] * = e α N j e ( t ) δ 0 k , k p j , k , k q ( t ) [ p j , k , k ( t ) ] * ,
ρ ˜ j e ( q , t ) = e α N j h ( t ) δ 0 k , k σ = x , y [ p j , k q , k σ ( t ) ] * p j , k , k σ ( t ) = e α N j h ( t ) δ 0 k , k [ p j , k q , k ( t ) ] * p j , k , k ( t ) .
Ω j , k , k x ( t ) = d cv d q q 2 + q 2 [ q ˜ j , x ( q , q , t ) + q ˜ j , x ( q , q , t ) ] | q = k k + k 1 k , k 1 k p j , k 1 , k 1 x ( t ) V k , k ; k 1 , k 1 eh ,
Ω j , k , k y ( t ) = d cv d q q 2 + q 2 [ q ˜ j , y ( q , q , t ) + q ˜ j , y ( q , q , t ) ] | q = k k + k 1 k , k 1 k p j , k 1 , k 1 y ( t ) V k , k ; k 1 , k 1 eh ,
˜ j ( q , q , t ) = d r e i q r g ( r ) d r e i q r ψ 0 e ( r R j ) ψ 0 h ( r R j ) E ( r , r , t ) ,
˜ j ( q , q , t ) = d r e i q r g ( r ) d r e i q r ψ 0 e ( r R j ) ψ 0 h ( r R j ) E ( r , r , t ) .
J ˜ j 1 D ( q , t ) = J ˜ j ( q , t ) ( e ^ w e ^ q ) = 1 i q ( e ^ w e ^ q ) t [ ρ ˜ j h ( q , t ) + ρ ˜ j e ( q , t ) ] = i q t [ ρ ˜ j h ( q , t ) + ρ ˜ j e ( q , t ) ] ,
𝕋 F ( Ω | ω 0 ) = d r | E ( r , r / 2 , Ω | ω 0 ) | 2 𝒲 ( 0 inc ) 2 ,
F ( Ω | ω 0 ) = + d r | E ( r , r / 2 , Ω | ω 0 ) | 2 𝒲 ( 0 inc ) 2 ,
H ( x , y , t = 0 ) = e ^ z H z 0 e i k 0 ( y y 0 ) exp { [ 1 + i b ( y y 0 ) ] x 2 x 2 ( y y 0 , L R ) } exp { [ 1 i a ( y y 0 ) ] ( y y 0 ) 2 y 2 ( y y 0 , L D ) } ,
A ˜ ( q , t = 0 ) = i ( q μ 0 q 2 ) × H ˜ ( q , t = 0 ) , E ˜ ( q , t = 0 ) = [ A ˜ ( q , t ) t ] t = 0 = ( q ω q μ 0 q 2 ) × H ˜ ( q , t = 0 ) .
ρ ^ e ( y ) = ψ ^ e ( y ) ψ ^ e ( y ) , ρ ^ h ( y ) = ψ ^ h ( y ) ψ ^ h ( y ) ,
ψ ^ e ( y ) = 1 k a ^ k e i k y , ψ ^ h ( y ) = 1 k β ^ k e i k y ,
ρ ˜ ^ e ( q ) = 1 k a ^ k q a ^ k , ρ ˜ ^ h ( q ) = 1 k β ˜ ^ ( k + q ) β ^ k .
n q e ( t ) = a ^ q a ^ q ,
n q h ( t ) = β ^ q β ^ ,
p q , q ( t ) = β ^ q a ^ q ,
p q , q * ( t ) = a ^ q β ^ q ,
{ a ^ k , a ^ k } = { β ^ k , β ^ k } = { a ^ k , β ^ k } = { a ^ k , β ^ k } = 0 , { a ^ k , a ^ k } = { β ^ k , β ^ k } = δ k , k .
a ^ k q a ^ k = a ^ k q { β ^ k , β ^ k } a ^ k = a ^ k q β ^ k β ^ k a ^ k + a ^ k q β ^ k β ^ k a ^ k = a ^ k q β ^ k β ^ k a ^ k a ^ k q β ^ k a ^ k β ^ k = a ^ k q β ^ k β ^ k a ^ k + a ^ k q a ^ k β ^ k β ^ k = a ^ k q β ^ k β ^ k a ^ k + a ^ k q a ^ k [ 1 β ^ k β ^ k ] .
a ^ k q a ^ k = a ^ k q β ^ k β ^ k a ^ k + a ^ k q a ^ k a ^ k q a ^ k β ^ k β ^ k a ^ k q β ^ k β ^ k a ^ k + a ^ k q a ^ k a ^ k q a ^ k β ^ k β ^ k p k q , k * ( t ) p k , k ( t ) + a ^ k q a ^ k [ 1 n k h ( t ) ] , a ^ k q a ^ k 2 N h ( t ) k p k q , k * ( t ) p k , k ( t ) ,
ρ ˜ e ( q , t ) 2 N h ( t ) k , k p k q , k * ( t ) p k , k ( t ) .
β ^ ( k + q ) β ^ k [ k 1 n e ( k 1 + q , t ) ] 1 k p k + q , k + q * ( t ) p k + q , k ( t ) = 2 N e ( t ) k p k , k + q * ( t ) p k , k ( t ) ,
ρ ˜ h ( q , t ) 2 N e ( t ) k , k p k , k q ( t ) p k , k * ( t ) .
V k 1 , k 1 ; k 2 , k 2 eh = β d 2 ξ d 2 ξ [ Ψ k 1 e ( ξ ) ] * [ Ψ k 1 h ( ξ ) ] * Ψ k 2 h ( ξ ) Ψ k 2 e ( ξ ) | ξ ξ |
V k 1 , k 2 ; k 3 , k 4 hh = β d 2 ξ d 2 ξ [ Ψ k 1 h ( ξ ) ] * [ Ψ k 2 h ( ξ ) ] * Ψ k 3 h ( ξ ) Ψ k 4 h ( ξ ) | ξ ξ |
V k 1 , k 2 ; k 3 , k 4 ee = β d 2 ξ d 2 ξ [ Ψ k 1 e ( ξ ) ] * [ Ψ k 2 e ( ξ ) ] * Ψ k 3 e ( ξ ) Ψ k 4 e ( ξ ) | ξ ξ |
V k 1 , k 1 ; k 2 , k 2 eh = δ k 1 + k 2 , k 1 + k 2 ( 2 β ) 𝒬 e , h ( k 1 k 2 ) ,
V k 1 , k 2 ; k 3 , k 4 hh = δ k 1 + k 2 , k 3 + k 4 ( 2 β ) 𝒬 h , h ( k 4 k 1 ) ,
V k 1 , k 2 ; k 3 , k 4 ee = δ k 1 + k 2 , k 3 + k 4 ( 2 β ) 𝒬 e , e ( k 1 k 4 ) ,
W j , k e , ( in ) ( t ) = 2 π k 1 | V k , k 1 ep | 2 n j , k 1 e ( t ) { N 0 ( Ω ph ) L ( ε k e ε k 1 e Ω ph , Γ ph ) + [ N 0 ( Ω ph ) + 1 ] L ( ε k e ε k 1 e Ω ph , Γ ph ) + 2 π k 1 k , k 1 | V k , k ; k 1 , k 1 eh | 2 [ 1 n j , k h ( t ) ] n j , k 1 h ( t ) n j , k 1 e ( t ) × L ( ε k e + ε k h ε k 1 e ε k 1 h , γ eh ) + 2 π k 2 , k 3 , k 4 | V k 1 , k 2 ; k 3 , k 4 ee | 2 [ 1 n j , k 2 e ( t ) ] n j , k 3 e ( t ) n j , k 4 e ( t ) × L ( ε k e + ε k 2 e ε k 3 e ε k 4 e , γ e ) ,
W j , k e , ( out ) ( t ) = 2 π k 1 | V k , k 1 ep | 2 [ 1 n j , k 1 e ( t ) ] { N 0 ( Ω ph ) L ( ε k 1 e ε k e Ω ph , Γ ph ) + [ N 0 ( Ω ph ) + 1 ] L ( ε k 1 e ε k e + Ω ph , Γ ph ) } + 2 π k 1 k , k 1 | V k 1 , k ; k 1 , k eh | 2 [ 1 n j , k h ( t ) ] n j , k 1 h ( t ) [ 1 n j , k 1 e ( t ) ] × L ( ε k 1 e + ε k h ε k e ε k 1 h , γ eh ) + 2 π k 2 , k 3 , k 4 | V k 4 , k 2 ; k 3 , k ee | 2 [ 1 n j , k 2 e ( t ) ] n j , k 3 e ( t ) [ 1 n j , k 4 e ( t ) ] × L ( ε k 4 e + ε k 2 e ε k 3 e ε k e , γ e ) ,
W j , k h , ( in ) ( t ) = 2 π k 1 | V k , k 1 hp | 2 n j , k 1 h ( t ) { N 0 ( Ω ph ) L ( ε k h ε k 1 h Ω ph , Γ ph ) + [ N 0 ( Ω ph ) + 1 ] L ( ε k h ε k 1 h + Ω ph , Γ ph ) } + 2 π k 1 k , k 1 | V k , k ; k 1 , k 1 eh | 2 [ 1 n j , k e ( t ) ] n j , k 1 e ( t ) n j , k 1 h ( t ) × L ( ε k e + ε k h ε k 1 e ε k 1 h , γ eh ) + 2 π k 2 , k 3 , k 4 | V k , k 2 ; k 3 , k 4 hh | 2 [ 1 n j , k 2 h ( t ) ] n j , k 3 h ( t ) n j , k 4 h ( t ) × L ( ε k h + ε k 2 h ε k 3 h ε k 4 h , γ h ) ,
W j , k h , ( out ) ( t ) = 2 π k 1 | V k , k 1 hp | 2 [ 1 n j , k 1 h ( t ) ] { N 0 ( Ω ph ) L ( ε k 1 h ε k h Ω ph , Γ ph ) + [ N 0 ( Ω ph ) + 1 ] L ( ε k 1 h ε k h + Ω ph , Γ ph ) } + 2 π k 1 k , k 1 | V k , k 1 ; k , k 1 eh | 2 [ 1 n j , k e ( t ) ] n j , k 1 e ( t ) × [ 1 n j , k 1 h ( t ) ] L ( ε k e + ε k 1 h ε k 1 e ε k h , γ eh ) + 2 π k 2 , k 3 , k 4 | V k 4 , k 2 ; k 3 , k hh | 2 [ 1 n j , k 2 h ( t ) ] n j , k 3 h ( t ) × [ 1 n j , k 4 h ( t ) ] L ( ε k 4 h + ε k 2 h ε k 3 h ε k h , γ h ) ,
| V k , k 1 hp | 2 = e 2 Ω ph 2 π 0 ( 1 1 s ) | 𝒬 h , h ( k 1 k ) | [ 1 D ( | k 1 k | , t ) ] 2 ,
| V k , k 1 ep | 2 = e 2 Ω ph 2 π 0 ( 1 1 s ) | 𝒬 e , e ( k 1 k ) | [ 1 D ( | k 1 k | , t ) ] 2 ,
Δ j , k e ( t ) = π k 1 , q 0 | V k 1 q , k + q ; k , k 1 ee | 2 [ L ( ε k 1 q e + ε k + q e ε k e ε k 1 e , γ e ) × { n j , k 1 q e ( t ) n j , k + q e ( t ) [ 1 n j , k 1 e ( t ) ] + [ 1 n j , k 1 q e ( t ) ] [ 1 n j , k + q e ( t ) ] n j , k 1 e ( t ) } ] + π k 1 , q 0 | V k q , k 1 q ; k 1 , k eh | 2 [ L ( ε k 1 q h + ε k q e ε k e ε k 1 h , γ eh ) { n j , k 1 q h ( t ) [ 1 n j , k 1 h ( t ) ] n j , k q e ( t ) + [ 1 n j , k 1 q h ( t ) ] n j , k 1 h ( t ) [ 1 n j , k q e ( t ) ] } ] ,
Δ j , k h ( t ) = π k 1 , q 0 | V k 1 q , k + q ; k , k 1 hh | 2 [ L ( ε k 1 q h + ε k + q h ε k h ε k 1 h , γ h ) × { n j , k 1 q h ( t ) n j , k + q h ( t ) [ 1 n j , k 1 h ( t ) ] + [ 1 n j , k 1 q h ( t ) ] [ 1 n j , k + q h ( t ) ] n j , k 1 h ( t ) } ] + π k 1 , q 0 | V k 1 q , k q ; k , k 1 eh | 2 [ L ( ε k 1 q e + ε k q h ε k h ε k 1 e , γ eh ) { n j , k 1 q e ( t ) [ 1 n j , k 1 e ( t ) ] × n j , k q h ( t ) + [ 1 n j , k 1 q e ( t ) ] n j , k 1 e ( t ) [ 1 n j , k q h ( t ) ] } ] .
Δ j , k , q e ( t ) = π k 1 | V k 1 , k + q ; k , k 1 + q ee | 2 [ L ( ε k 1 + q e + ε k e ε k 1 e ε k + q e , γ e ) × { n j , k 1 + q e ( t ) n j , k e ( t ) [ 1 n j , k 1 e ( t ) ] + [ 1 n j , k 1 + q e ( t ) ] [ 1 n j , k e ( t ) ] n j , k 1 e ( t ) } ] + π k 1 | V k , k 1 q ; k 1 , k + q eh | 2 [ L ( ε k 1 q h + ε k e ε k 1 h ε k + q e , γ eh ) { n j , k 1 q h ( t ) [ 1 n j , k 1 h ( t ) ] n j , k e ( t ) + [ 1 n j , k 1 q h ( t ) ] n j , k 1 h ( t ) [ 1 n j , k q e ( t ) ] } ] ,
Δ j , k , q h ( t ) = π k 1 | V k 1 , k + q ; k , k 1 + q ee | 2 [ L ( ε k 1 + q h + ε k h ε k 1 h ε k + q h , γ h ) × { n j , k 1 + q h ( t ) n j , k h ( t ) [ 1 n j , k 1 h ( t ) ] + [ 1 n j , k 1 + q h ( t ) ] [ 1 n j , k h ( t ) ] n j , k 1 h ( t ) } ] + π k 1 | V k 1 , k q ; k , k 1 + q eh | 2 [ L ( ε k 1 q e + ε k h ε k 1 e ε k + q h , γ eh ) { n j , k 1 q e ( t ) [ 1 n j , k 1 e ( t ) ] × n j , k h ( t ) + [ 1 n j , k 1 q e ( t ) ] n j , k 1 e ( t ) [ 1 n j , k h ( t ) ] } ] .

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