Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations

Journal name:
Nature Photonics
Year published:
Published online


Ultrafast charge transport in strongly biased semiconductors is at the heart of high-speed electronics, electro-optics and fundamental solid-state physics1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13. Intense light pulses in the terahertz spectral range have opened fascinating vistas14, 15, 16, 17, 18, 19, 20, 21. Because terahertz photon energies are far below typical electronic interband resonances, a stable electromagnetic waveform may serve as a precisely adjustable bias5, 11, 17, 19. Novel quantum phenomena have been anticipated for terahertz amplitudes, reaching atomic field strengths8, 9, 10. We exploit controlled (multi-)terahertz waveforms with peak fields of 72 MV cm−1 to drive coherent interband polarization combined with dynamical Bloch oscillations in semiconducting gallium selenide. These dynamics entail the emission of phase-stable high-harmonic transients, covering the entire terahertz-to-visible spectral domain between 0.1 and 675 THz. Quantum interference of different ionization paths of accelerated charge carriers is controlled via the waveform of the driving field and explained by a quantum theory of inter- and intraband dynamics. Our results pave the way towards all-coherent terahertz-rate electronics.

At a glance


  1. Field-sensitive terahertz nonlinear optics.
    Figure 1: Field-sensitive terahertz nonlinear optics.

    a, The waveform of the terahertz driving field (blue, solid curve) features a Gaussian envelope (black dashed curve) with an intensity full-width at half-maximum of 109 fs, which contains three optical cycles. The transient was recorded electro-optically in a GaSe sensor (thickness, 40 µm), with an 8 fs near-infrared gate pulse (centre wavelength, 0.84 µm). Inset: corresponding amplitude spectrum. b, Electro-optic trace of the waveform generated by the intense terahertz pulse of a in a GaSe single crystal (thickness, 220 µm; angle of incidence, θ = 70°), kept at room temperature. The data are shown as recorded with an AgGaS2 detector (thickness, 100 µm), not corrected for the detector response, which leads to a temporal walk-off between the fundamental and the second harmonic. A superposition of the fundamental, second harmonic and optical rectification components is clearly seen. Insets: corresponding amplitude spectrum and experimental geometry, indicating the angle of incidence θ. The role of in-plane rotation of GaSe (angle ϕ) is investigated in Supplementary Fig. 5.

  2. CEP-stable terahertz HHG in bulk GaSe.
    Figure 2: CEP-stable terahertz HHG in bulk GaSe.

    a, HH intensity spectrum (solid line and shaded area) emitted from a GaSe single crystal (thickness, 220 µm; θ = 50°) driven by the phase-locked terahertz pulse in Fig. 1a. A combination of electro-optic sampling (EOS), an InGaAs diode array and a silicon CCD maps out the HH spectrum throughout the terahertz, far-infrared, mid-infrared, near-infrared and visible regimes. Eg/h marks the bandgap frequency (476 THz). The blue dashed curve shows the computed HH intensity spectrum, obtained from a five-band model (see Chapter 3 in the Supplementary Information for details). b, Dependence of the intensity I13 of the 13th harmonic on the incident terahertz amplitude Ea (top scale). Bottom scale: internal terahertz amplitude Eint, obtained from Ea by accounting for reflection at the crystal surface. Symbols, experimental data; dashed line, scaling law I13  Ea26; dotted line, scaling law I13  Ea. c, The spectral interference between the frequency-doubled sixth harmonic and the 12th harmonic confirms the CEP stability of the HH radiation. For experimental details of f − 2f interferometry, see Supplementary Fig. 2. d, Model electronic band structure of GaSe between the Γ- and K-points, underlying the subsequent computations. Two conduction bands (CB1 and CB2) and three valence bands (VB1, VB2 and VB3) are taken into account. Coherent excitation of electrons (symbols), for example, from the second valence to the lowest conduction band, can proceed via interfering pathways with different scaling in powers of the terahertz field.

  3. Sub-cycle carrier dynamics driven by CEP-stable terahertz transients.
    Figure 3: Sub-cycle carrier dynamics driven by CEP-stable terahertz transients.

    a, Phase-locked terahertz waveform driving coherent interband excitation and carrier transport. The calculated temporal dynamics of the distribution of electrons in the first conduction band, ne = fke1, is shown as colour maps, for peak electric fields Eint of 8 MV cm−1 (b), 11 MV cm−1 (c) and 14 MV cm−1 (d). When the electron's wavevector k reaches the Brillouin zone boundary at the K-point, Bragg reflection occurs and the sign of k is instantly inverted. White lines trace the centre of the electron distribution for different delay times.

  4. CEP control of terahertz HHG in GaSe.
    Figure 4: CEP control of terahertz HHG in GaSe.

    a, HH intensity spectra measured for a CEP of the driving field of ϕCEP = 0.1π (blue) and 1.1π (red). We define ϕCEP to be zero when a positive field maximum of the carrier wave coincides with the peak of the Gaussian envelope. Inset: corresponding waveforms of the driving fields. b, Systematic dependence of HH spectra on the CEP of the terahertz transient. Although HHG depends only moderately on the CEP for orders n < 15, CEP variation leads to a pronounced frequency shift for n ≥ 15 with a slope of −2.5 THz rad−1 (indicated by the dashed lines). c, CEP dependence of the HH spectra computed via the quantum five-band model.


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Author information


  1. Department of Physics, University of Regensburg, 93040 Regensburg, Germany

    • O. Schubert,
    • M. Hohenleutner,
    • F. Langer,
    • B. Urbanek,
    • C. Lange &
    • R. Huber
  2. Department of Physics, University of Marburg, 35032 Marburg, Germany

    • U. Huttner,
    • D. Golde,
    • M. Kira &
    • S. W. Koch
  3. Department of Physics, University of Paderborn, 33098 Paderborn, Germany

    • T. Meier


O.S., M.H., F.L. and R.H. conceived the study. O.S., M.H., F.L., B.U., C.L. and R.H. carried out the experiment. U.H., D.G., T.M., M.K. and S.W.K. developed the quantum-mechanical model and carried out the computations. O.S., M.H., F.L., U.H., M.K., S.W.K and R.H. wrote the manuscript. All authors discussed the results.

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