Acceleration and collision of particles has been a key strategy for exploring the texture of matter. Strong light waves can control and recollide electronic wavepackets, generating high-harmonic radiation that encodes the structure and dynamics of atoms and molecules and lays the foundations of attosecond science1,2,3. The recent discovery of high-harmonic generation in bulk solids4,5,6 combines the idea of ultrafast acceleration with complex condensed matter systems, and provides hope for compact solid-state attosecond sources6,7,8 and electronics at optical frequencies3,5,9,10. Yet the underlying quantum motion has not so far been observable in real time. Here we study high-harmonic generation in a bulk solid directly in the time domain, and reveal a new kind of strong-field excitation in the crystal. Unlike established atomic sources1,2,3,9,11, our solid emits high-harmonic radiation as a sequence of subcycle bursts that coincide temporally with the field crests of one polarity of the driving terahertz waveform. We show that these features are characteristic of a non-perturbative quantum interference process that involves electrons from multiple valence bands. These results identify key mechanisms for future solid-state attosecond sources and next-generation light-wave electronics. The new quantum interference process justifies the hope for all-optical band-structure reconstruction and lays the foundation for possible quantum logic operations at optical clock rates.
This is a preview of subscription content
Subscribe to Journal
Get full journal access for 1 year
only $3.90 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Tax calculation will be finalised during checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Paul, P. M. et al. Observation of a train of attosecond pulses from high harmonic generation. Science 292, 1689–1692 (2001)
Dudovich, N. et al. Measuring and controlling the birth of attosecond XUV pulses. Nature Phys. 2, 781–786 (2006)
Krausz, F. & Stockman, M. I. Attosecond metrology: from electron capture to future signal processing. Nature Photon. 8, 205–213 (2014)
Ghimire, S. et al. Observation of high-order harmonic generation in a bulk crystal. Nature Phys. 7, 138–141 (2011)
Schubert, O. et al. Sub-cycle control of terahertz high-harmonic generation by dynamical Bloch oscillations. Nature Photon. 8, 119–123 (2014)
Ghimire, S. et al. Strong-field and attosecond physics in solids. J. Phys. B 47, 204030 (2014)
Higuchi, T., Stockman, M. I. & Hommelhoff, P. Strong-field perspective on high-harmonic radiation from bulk solids. Phys. Rev. Lett. 113, 213901 (2014)
Mücke, O. D. Isolated high-order harmonics pulse from two-color-driven Bloch oscillations in bulk semiconductors. Phys. Rev. B 84, 081202 (2011)
Goulielmakis, E. et al. Attosecond control and measurement: lightwave electronics. Science 317, 769–775 (2007)
Zaks, B., Liu, R. B. & Sherwin, M. S. Experimental observation of electron–hole recollisions. Nature 483, 580–583 (2012)
Shafir, D. et al. Resolving the time when an electron exits a tunnelling barrier. Nature 485, 343–346 (2012)
Drescher, M. et al. Time-resolved atomic inner-shell spectroscopy. Nature 419, 803–807 (2002)
Calegari, F. et al. Ultrafast electron dynamics in phenylalanine initiated by attosecond pulses. Science 346, 336–339 (2014)
Neppl, S. et al. Direct observation of electron propagation and dielectric screening on the atomic length scale. Nature 517, 342–346 (2015)
Smirnova, O. et al. High harmonic interferometry of multi-electron dynamics in molecules. Nature 460, 972–977 (2009)
Salières, P. et al. Feynman’s path-integral approach for intense-laser-atom interactions. Science 292, 902–905 (2001)
Corkum, P. B. Plasma perspective on strong-field multiphoton ionization. Phys. Rev. Lett. 71, 1994–1997 (1993)
Shafir, D., Mairesse, Y., Villeneuve, D. M., Corkum, P. B. & Dudovich, N. Atomic wavefunctions probed through strong-field light-matter interaction. Nature Phys. 5, 412–416 (2009)
Kanai, T., Minemoto, S. & Sakai, H. Quantum interference during high-order harmonic generation from aligned molecules. Nature 435, 470–474 (2005)
Golde, D., Meier, T. & Koch, S. W. High harmonics generated in semiconductor nanostructures by the coupled dynamics of optical inter- and intraband excitations. Phys. Rev. B 77, 075330 (2008)
Földi, P., Benedict, M. G. & Yakovlev, V. S. The effect of dynamical Bloch oscillations on optical-field-induced current in a wide-gap dielectric. New J. Phys. 15, 063019 (2013)
Vampa, G. et al. Theoretical analysis of high-harmonic generation in solids. Phys. Rev. Lett. 113, 073901 (2014)
Hawkins, P. G., Ivanov, M. Y. & Yakovlev, V. S. Effect of multiple conduction bands on high-harmonic emission from dielectrics. Phys. Rev. A 91, 013405 (2015)
Chin, A. H., Calderón, O. G. & Kono, J. Extreme midinfrared nonlinear optics in semiconductors. Phys. Rev. Lett. 86, 3292–3295 (2001)
Ghimire, S. et al. Generation and propagation of high-order harmonics in crystals. Phys. Rev. A 85, 043836 (2012)
Sekikawa, T., Katsura, T., Miura, S. & Watanabe, S. Measurement of the intensity-dependent atomic dipole phase of a high harmonic by frequency resolved optical gating. Phys. Rev. Lett. 88, 193902 (2002)
Kemper, A. F., Moritz, B., Freericks, J. K. & Devereaux, T. P. Theoretical description of high-order harmonic generation in solids. New J. Phys. 15, 023003 (2013)
Zener, C. A. Theory of the electrical breakdown of solid dielectrics. Proc. R. Soc. Lond. A 145, 523–529 (1934)
Zhao, H., Loren, E. J., van Driel, H. M. & Smirl, A. L. Coherence control of Hall charge and spin currents. Phys. Rev. Lett. 96, 246601 (2006)
Ladd, T. D. et al. Quantum computers. Nature 464, 45–53 (2010)
Sell, A., Leitenstorfer, A. & Huber, R. Phase-locked generation and field-resolved detection of widely tunable terahertz pulses with amplitudes exceeding 100 MV/cm. Opt. Lett. 33, 2767–2769 (2008)
Eimerl, D., Davis, L., Velsko, S., Graham, E. K. & Zalkin, A. Optical, mechanical, and thermal properties of barium borate. J. Appl. Phys. 62, 1968–1983 (1987)
Ruffin, A. B., Rudd, J. V., Whitaker, J. F., Feng, S. & Winful, H. G. Direct observation of the Gouy phase shift with single-cycle terahertz pulses. Phys. Rev. Lett. 83, 3410–3413 (1999)
Gallot, G. & Grischkowsky, D. Electro-optic detection of terahertz radiation. J. Opt. Soc. Am. B 16, 1204–1212 (1999)
Linden, S., Giessen, H. & Kuhl, J. XFROG — A new method for amplitude and phase characterization of weak ultrashort pulses. Phys. Status Solidi B 206, 119–124 (1998)
Kane, D. J. Real-time measurement of ultrashort laser pulses using principal component generalized projections. IEEE J. Sel. Top. Quantum Electron. 4, 278–284 (1998)
Wyatt, A. Frequency-resolved optical gating. http://www.mathworks.com/matlabcentral/fileexchange/16235-frequency-resolved-optical-gating-frog- (MATLAB central file exchange, 7 July 2008)
Kane, D. J. Recent progress toward real-time measurement of ultrashort laser pulses. IEEE J. Quantum Electron. 35, 421–431 (1999)
Golde, D., Kira, M., Meier, T. & Koch, S. W. Microscopic theory of the extremely nonlinear terahertz response of semiconductors. Phys. Status Solidi B 248, 863–866 (2011)
Schlüter, M. et al. Optical properties of GaSe and GaSxSe1 - x mixed crystals. Phys. Rev. B 13, 3534–3547 (1976)
Segura, A., Bouvier, J., Andrés, M. V., Manjón, F. J. & Muñoz, V. Strong optical nonlinearities in gallium and indium selenides related to inter-valence-band transitions induced by light pulses. Phys. Rev. B 56, 4075–4084 (1997)
Moiseyev, N. Selection rules for harmonic generation in solids. Phys. Rev. A 91, 053811 (2015)
Kira, M. & Koch, S. W. Semiconductor Quantum Optics (Cambridge Univ. Press, 2011)
Golde, D., Meier, T. & Koch, S. W. Microscopic analysis of extreme nonlinear optics in semiconductor nanostructures. J. Opt. Soc. Am. B 23, 2559–2565 (2006)
Schultze, M. et al. Attosecond band-gap dynamics in silicon. Science 346, 1348–1352 (2014)
Vu, Q. T. et al. Light-induced gaps in semiconductor band-to-band transitions. Phys. Rev. Lett. 92, 217403 (2004)
Schuh, K., Hader, J., Moloney, J. V. & Koch, S. W. Influence of many-body interactions during the ionization of gases by short intense optical pulses. Phys. Rev. E 89, 033103 (2014)
Kira, M., Jahnke, F., Hoyer, W. & Koch, S. W. Quantum theory of spontaneous emission and coherent effects in semiconductor microstructures. Prog. Quantum Electron. 23, 189–279 (1999)
Liu, Y.-W. in Fourier Transform Applications (ed. Salih, S. ) 291–300 (InTech, 2012)
The work in Regensburg was supported by the European Research Council through grant no. 305003 (QUANTUMsubCYCLE) as well as by the Deutsche Forschungsgemeinschaft (through grant number HU 1598/2-1) and the work in Marburg by the Deutsche Forschungsgemeinschaft (through SFB 1083 and grant number KI 917/2-2).
The authors declare no competing financial interests.
Extended data figures and tables
Extended Data Figure 1 High-order harmonic (HH) spectrum generated by intense phase-locked terahertz pulses in bulk GaSe.
The ultrabroadband HH emission is recorded using a monochromator with a calibrated pyroelectric detector (PED), a lead sulfide diode (PbS) and spectrometers employing InGaAs and cooled Si detectors. At a frequency of 476 THz interband photoluminescence (PL) dominates the spectrum. The driving multi-terahertz waveform is shown in the inset.
a, Electro-optic signal of the multi-terahertz driving field as recorded using a BBO detector (thickness, 10 µm; blue curve, SEOS-BBO) and a 〈110〉-oriented ZnTe crystal (thickness, 6.5 µm; black dashed curve, SEOS-ZnTe) for spectral components between 12 and 45 THz. The black solid curve represents the corresponding terahertz electric field EDet as a function of delay time after correction for the complex-valued detector response of the ZnTe crystal displayed in b. b, Absolute value (black solid curve) and phase (red dashed curve) of the transfer function34 calculated for a 6.5-µm-thick ZnTe electro-optic detector in the multi-terahertz frequency range.
a, Retrieved spectral intensities of the detected HH pulse sequence (grey, numerals indicate harmonic orders), the gating pulse (blue) and the sum-frequency intensity (SF) for optimal temporal overlap of gating and HH pulses (black dashed). All spectra displayed here correspond to the data set of main text Fig. 1. The measured spectral intensity of the sum-frequency signal is shown as a red shaded area for comparison. All spectra are normalized to their individual maximum and shifted in intensity for clarity. b, Temporal intensity profiles of the gating pulse as measured and reconstructed via SHG-FROG (blue) and double-blind XFROG (black dashed).
a, Tight binding band structure of GaSe with three valence bands (red) and two conduction bands (blue). The bandgap EG is marked by the black arrow. b, Schematics of interband transitions. The external electric field (orange curve) stimulates a transition between electronic states (blue spheres) in different bands via the dipole moment , indicated by the black arrow. c, Schematics depicting intraband currents. Carriers (blue spheres) are accelerated by the external electric field (orange curve) inside their respective bands.
a, Simplified band-structure schematics consisting of two valence bands h1 and h2 (red lines) and one conduction band e (blue line). The allowed transitions are labelled and marked by black arrows. b, Schematic of the direct excitation path. The transition h1 → e is marked by the blue arrow. c, Indirect excitation path consisting of the transition h1 → h2 (red arrow) and h2 → e (three-dimensional red arrow). d, Full quantum interference (QI): combination of the direct excitation path h1 → e (blue arrow) and the indirect excitation path h1 → h2 → e (red arrows). e, The interference path efficiency ηIPE(ETHz), black line, describes the fraction of additional carriers promoted to the conduction band e1 by indirect paths such as h1 → h2 → e as a function of the driving terahertz field strength. For the peak fields used in our calculations (indicated by dashed red lines), the strength of indirect excitations is of the same order of magnitude as the strength of direct excitations .
Globally normalized HH intensity envelopes computed within the five-band model as a function of delay time and the coherent control factor Fcc regulating coherent transitions between occupied valence bands. The same data are presented in Fig. 2c in normalized form. Bright colours mark strong emission, dark colours mark weaker emission (colour key at top right).
The black curve depicts the retrieved relative phase of typical HH bursts for driving peak fields of 33 MV cm−1. A trivial linear contribution to the phase representing the weighted central frequency νc of detected HHs within the detection bandwidth has been subtracted for better visibility: . The intensity envelope IHH (blue shaded area) and the derived instantaneous frequency νi of the three main HH pulses (red curves) are shown on the same timescale.
a, False-colour plot of the reconstructed intensity envelopes of the HH pulse sequence versus delay time t for different carrier-envelope phases ϕCEP of the driving waveform. b, Reconstructed HH intensity envelopes for ϕCEP = 0 (black) and ϕCEP = π (red).
Intensity envelope (blue shaded area) of spectrally filtered HH bursts (inset, blue shaded spectrum) emitted from GaSe featuring a full-width at half-maximum of 3 fs only, as calculated with a semiclassical intraband model. Inset, calculated amplitude spectrum driven by a multi-terahertz transient with external peak fields of 43 MV cm−1 as emitted from the sample (grey shaded) and spectrally filtered (blue shaded) with a suitable high-pass filter (black dashed).
a, Schematic of a triangle system of electronic states as a minimal condition for non-perturbative quantum interference. Filled circles 1 and 2 denote occupied states, whereas the sphere e symbolises an empty state. b, A symmetric potential (black solid line) and the resulting wave functions with even (red shaded area) and odd (blue shaded area) parity. c, System with an asymmetric potential (black line) resulting in wave functions with indefinite parity (orange shaded areas). See Methods for details.
About this article
Cite this article
Hohenleutner, M., Langer, F., Schubert, O. et al. Real-time observation of interfering crystal electrons in high-harmonic generation. Nature 523, 572–575 (2015). https://doi.org/10.1038/nature14652
Light-wave control of correlated materials using quantum magnetism during time-periodic modulation of coherent transport
Communications Physics (2021)
Nature Physics (2021)
Optical Review (2021)
Nature Photonics (2020)