Abstract

In this paper, a near-infrared (NIR) laser (with frequency ωNIR) exciting and an intense mid-infrared (MIR) bichromatic laser (with frequencies ω1 and ω2) driving scheme is proposed in semiconductors. It is predicted that if the frequency difference between the two components of the MIR laser, i.e. Ω = ω1-ω2, is in dozens of GHz frequency range, the emitting light contains spectra with frequencies ω = ωNIR + NΩ, where N is integer. In analogy with high-order THz sideband generation (HSG) in semiconductors, this phenomenon is named difference frequency sideband generation (DSG). Similar to the HSG case, the emitted sideband spectrum in DSG exhibits a nonperturbative plateau, where the intensity of the sideband remains approximately constant up to a cutoff frequency. The location of the cutoff frequency and its relationship with the frequency detuning of the NIR laser are discussed via the semi-classical saddle-point method.

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

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  1. K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
    [Crossref]
  2. K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
    [Crossref]
  3. B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
    [Crossref]
  4. S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
    [Crossref] [PubMed]
  5. G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
    [Crossref] [PubMed]
  6. S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
    [Crossref]
  7. T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
    [Crossref]
  8. R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
    [Crossref]
  9. J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
    [Crossref]
  10. J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  18. F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
    [Crossref]
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    [Crossref]
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2017 (3)

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
[Crossref]

2015 (1)

2014 (4)

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

2013 (3)

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

2012 (2)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

2011 (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

2010 (1)

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

2008 (1)

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
[Crossref]

2007 (1)

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

2005 (1)

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

1998 (1)

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Agostini, P.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Akiyama, H.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Allen, S. J.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Banks, H.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

Birkedal, V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Birnir, B.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Brabec, T.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Carter, S. G.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Chatterjee, S.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Citrin, D. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Coldren, L. A.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Corkum, P. B.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Crosse, J. A.

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

DiChiara, A. D.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

DiMauro, L. F.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Ewers, B.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Ghimire, S.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Gibbs, H. M.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Gossard, A. C.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Hirori, H.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Huang, T.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Jauho, A.-P.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Johnsen, K.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Kadoya, Y.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Khitrova, G.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Kira, M.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Klettke, A. C.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Klug, D. D.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Koch, M.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Koch, S. W.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Kono, J.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Koster, N. S.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Lan, P.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Li, L.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Liu, H.

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

H. Liu and W. She, “Theory of effect of near-infrared laser polarization direction on high-order terahertz sideband generation in semiconductors,” Opt. Express 23(8), 10680–10686 (2015).
[Crossref] [PubMed]

Liu, R.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Liu, R. B.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Liu, R.-B.

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Liu, X.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Lu, P.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Mack, S.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Maslov, A. V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

McDonald, C. R.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Nagai, M.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Noda, T.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Nordstrom, K. B.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Orlando, G.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Reis, D. A.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Sakaki, H.

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

Satoh, N.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

She, W.

Sherwin, M. S.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Shinokita, K.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Sistrunk, E.

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Tanaka, K.

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

Vampa, G.

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

Wang, C. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Woscholski, R.

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Xu, X.

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

Yan, J.-Y.

J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
[Crossref]

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
[Crossref]

Yang, F.

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Zaks, B.

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

Zhang, X.

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

Zhu, B.-F.

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Zhu, X.

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

AIP Conf. Proc. (1)

R.-B. Liu and B.-F. Zhu, “High-order THz-sideband generation in semiconductors,” AIP Conf. Proc. 893, 1455–1456 (2007).
[Crossref]

Appl. Phys. Lett. (2)

K. Shinokita, H. Hirori, M. Nagai, N. Satoh, Y. Kadoya, and K. Tanaka, “Dynamical Franz–Keldysh effect in GaAs/AlGaAs multiple quantum wells induced by single-cycle terahertz pulses,” Appl. Phys. Lett. 97(21), 211902 (2010).
[Crossref]

B. Zaks, H. Banks, and M. S. Sherwin, “High-order sideband generation in bulk GaAs,” Appl. Phys. Lett. 102(1), 012104 (2013).
[Crossref]

J. Appl. Phys. (1)

J.-Y. Yan, “High-order sideband generation in a semiconductor quantum well driven by two orthogonal terahertz fields,” J. Appl. Phys. 122(8), 084306 (2017).
[Crossref]

Nat. Commun. (1)

J. A. Crosse, X. Xu, M. S. Sherwin, and R. B. Liu, “Theory of low-power ultra-broadband terahertz sideband generation in bi-layer graphene,” Nat. Commun. 5(1), 4854 (2014).
[Crossref] [PubMed]

Nat. Phys. (1)

S. Ghimire, A. D. DiChiara, E. Sistrunk, P. Agostini, L. F. DiMauro, and D. A. Reis, “Observation of high-order harmonic generation in a bulk crystal,” Nat. Phys. 7(2), 138–141 (2011).
[Crossref]

Nature (1)

B. Zaks, R. B. Liu, and M. S. Sherwin, “Experimental observation of electron-hole recollisions,” Nature 483(7391), 580–583 (2012).
[Crossref] [PubMed]

New J. Phys. (2)

F. Yang and R.-B. Liu, “Berry phases of quantum trajectories of optically excited electron–hole pairs in semiconductors under strong terahertz fields,” New J. Phys. 15(11), 115005 (2013).
[Crossref]

F. Yang, X. Xu, and R.-B. Liu, “Giant Faraday rotation induced by the Berry phase in bilayer graphene under strong terahertz fields,” New J. Phys. 16(4), 043014 (2014).
[Crossref]

Opt. Commun. (1)

H. Liu and X. Zhang, “Theory of controlling band-width broadening in terahertz sideband generation in semiconductors by a direct current electric field,” Opt. Commun. 387, 37–42 (2017).
[Crossref]

Opt. Express (1)

Phys. Rev. A (Coll. Park) (1)

T. Huang, X. Zhu, L. Li, X. Liu, P. Lan, and P. Lu, “High-order-harmonic generation of a doped semiconductor,” Phys. Rev. A (Coll. Park) 96(4), 043425 (2017).
[Crossref]

Phys. Rev. B Condens. Matter Mater. Phys. (3)

J.-Y. Yan, “Theory of excitonic high-order sideband generation in semiconductors under a strong terahertz field,” Phys. Rev. B Condens. Matter Mater. Phys. 78(7), 075204 (2008).
[Crossref]

J. A. Crosse and R.-B. Liu, “Quantum-coherence-induced second plateau in high-sideband generation,” Phys. Rev. B Condens. Matter Mater. Phys. 89(12), 121202 (2014).
[Crossref]

B. Ewers, N. S. Koster, R. Woscholski, M. Koch, S. Chatterjee, G. Khitrova, H. M. Gibbs, A. C. Klettke, M. Kira, and S. W. Koch, “Ionization of coherent excitons by strong terahertz fields,” Phys. Rev. B Condens. Matter Mater. Phys. 85(7), 075307 (2012).
[Crossref]

Phys. Rev. Lett. (3)

K. B. Nordstrom, K. Johnsen, S. J. Allen, A.-P. Jauho, B. Birnir, J. Kono, T. Noda, H. Akiyama, and H. Sakaki, “Excitonic Dynamical Franz-Keldysh Effect,” Phys. Rev. Lett. 81(2), 457–460 (1998).
[Crossref]

G. Vampa, C. R. McDonald, G. Orlando, D. D. Klug, P. B. Corkum, and T. Brabec, “Theoretical analysis of high-harmonic generation in solids,” Phys. Rev. Lett. 113(7), 073901 (2014).
[Crossref] [PubMed]

H. Banks, B. Zaks, F. Yang, S. Mack, A. C. Gossard, R. Liu, and M. S. Sherwin, “Terahertz electron-hole recollisions in GaAs/AlGaAs quantum wells: robustness to scattering by optical phonons and thermal fluctuations,” Phys. Rev. Lett. 111(26), 267402 (2013).
[Crossref] [PubMed]

Science (1)

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310(5748), 651–653 (2005).
[Crossref] [PubMed]

Other (1)

H. Haug and S. W. Koch, Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed. (World Scientific Publishing Company, 2009).

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Figures (4)

Fig. 1
Fig. 1 The schematic diagram of the frequency spectrum of the DSG.
Fig. 2
Fig. 2 The intensity of the sideband spectrum. The black arrow gives the position of the cutoff frequency of the spectrum plateau, which is N max Ω=4 U p .
Fig. 3
Fig. 3 The dimensionless sideband frequency shift N Ω Ω/ U p as a function of the delay time for various frequency detuning △. Where (a) △ = 0; (b) △ = 0.5Up; (c) △ = Up and (d) △ = 1.5Up. The red and blue curves are the results of N Ω Ω=[ 1cos( Ωτ ) ]( 2 U p Δ )+sin( Ωτ ) 4 U p Δ Δ 2 and N Ω Ω=[ 1cos( Ωτ ) ]( 2 U p Δ )sin( Ωτ ) 4 U p Δ Δ 2 respectively, which coincide with each other in (a).
Fig. 4
Fig. 4 (a) The intensity of the sideband spectrum under △ = 0.5Up, Up and 1.5Up. (b) The intensity of the sideband spectrum as a function of △ and the sideband frequency. The black and red dotted lines in (b) are respective the upper and lower cutoffs extracted through Eq. (15). The calculation conditions are same as that of Fig. 2 except △.

Equations (23)

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F( t )= F 1 e iθ cos( ω 1 t )+ F 2 cos( ω 2 t ),
F( t )= F 0 ( t )cos( ω mod t )=2Fcos( ω a t )cos( ω mod t )=Fcos( ω 1 t )+Fcos( ω 2 t ).
i ρ cv ( k,t ) t ={ 1 2 m cv [ keA( t ) ] 2 + E g i γ 2 } ρ cv ( k,t )+ d cv F NIR ( t ),
ρ cv ( k,t )= i 0 e iS( k,t,τ )/ iωt γ 2 τ d cv F NIR dτ,
S( k,t,τ )= tτ t [ keA( t ) ] 2 2 m cv d t ( ω NIR E g )τ( ω ω NIR )t,
P( t )= 1 ( 2π ) 3 d cv ρ cv ( k,t ) d 3 k
P( t )= i d cv d cv F NIR ( 2π ) 3 0 2π m cv iτ 3 e i( ω NIR E g / )τi ω NIR t γ 2 τ e iκ( τ ) , × e i ξ 1 ( τ )cos( 2 ω 1 t ω 1 τ ) e i ξ 2 ( τ )cos( 2 ω 2 t ω 2 τ ) e i ξ a ( τ )cos( 2 ω a t ω a τ ) e i ξ Ω ( τ )cos( Ωt Ωτ/2 ) dτ
κ( τ )= e 2 F 2 2 m cv τ [ 1cos( ω 1 τ ) ω 1 4 + 1cos( ω 2 τ ) ω 2 4 ] e 2 F 2 τ 4 m cv { 1 ω 1 2 + 1 ω 2 2 },
ξ a ( τ )= e 2 F 2 m cv ω 1 ω 2 { 1 2 ω a sin( ω a τ ) 2 ω 1 ω 2 τ sin( ω 2 τ 2 )sin( ω 1 τ 2 ) },
ξ 1 ( τ )= e 2 F 2 2 m cv ω 1 3 { 1 2 sin( ω 1 τ ) 1 ω 1 τ [ 1cos( ω 1 τ ) ] },
ξ 2 ( τ )= e 2 F 2 2 m cv ω 2 3 { 1 2 sin( ω 2 τ ) 1 ω 2 τ [ 1cos( ω 2 τ ) ] },
ξ Ω ( τ )= e 2 F 2 m cv ω 1 ω 2 { 2 ω 1 ω 2 τ sin( ω 2 τ 2 )sin( ω 1 τ 2 ) 1 Ω sin( Ωτ 2 ) }.
P( ω NIR + N a 2 ω a + N 1 2 ω 1 + N 2 2 ω 2 + N Ω Ω )= i d cv d cv F NIR ( 2π ) 3 0 2π m cv iτ 3 e i( ω NIR E g / )τ γ 2 τ e iκ( τ )    × i N a N 1 N 2 N Ω J N a { ξ a ( τ ) } J N 1 { ξ 1 ( τ ) } J N 2 { ξ 2 ( τ ) } J N Ω { ξ Ω ( τ ) } e i( N a ω a + N 1 ω 1 + N 2 ω 2 + N Ω Ω/2 )τ dτ
P( ω NIR + N Ω Ω )= i d cv d cv F NIR ( 2π ) 3 0 2π m cv iτ 3 e i( ω NIR E g / )τ γ 2 τ e iκ( τ )    × i N Ω J 0 { ξ a ( τ ) } J 0 { ξ 1 ( τ ) } J 0 { ξ 2 ( τ ) } J N Ω { ξ Ω ( τ ) } e i N Ω Ωτ/2 dτ
k S( k,t,τ )=0k= e τ tτ t A( t )d t .
S( t,τ )=κ( τ ) ξ Ω ( τ )cos( Ωt Ωτ/2 ) ξ 1 ( τ )cos( 2 ω 1 t ω 1 τ )                ξ 2 ( τ )cos( 2 ω 2 t ω 2 τ ) ξ 0 ( τ )cos( 2 ω a t ω a τ )Δτ                N a 2 ω a t N 1 2 ω 1 t N 2 2 ω 2 t N Ω Ωt
S( t,τ )=κ( τ ) ξ Ω ( τ )cos( Ωt Ωτ/2 )Δτ N Ω Ωt
κ( τ ) e 2 F 2 τ 4 m cv { 1 ω 1 2 + 1 ω 2 2 } 2 U p τ .
ξ Ω ( τ ) e 2 F 2 m cv ω 1 ω 2 Ω sin( Ωτ 2 ) 4 U p Ω sin( Ωτ 2 ).
S( t,τ )=2 U p τ+ 4 U p Ω sin( Ωτ 2 )cos( Ωt Ωτ/2 )Δτ N Ω Ωt.
τ S( t,τ )=0cos( Ωτ 2 )cos( Ωt Ωτ 2 )+sin( Ωτ 2 )sin( Ωt Ωτ 2 )= Δ2 U p 2 U p ,
t S( t,τ )=0 N Ω Ω=4 U p sin( Ωτ 2 )sin( Ωt Ωτ/2 )
N Ω Ω=[ 1cos( Ωτ ) ]( 2 U p Δ )±sin( Ωτ ) 4 U p Δ Δ 2

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