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# Real-time observation of interfering crystal electrons in high-harmonic generation

## Abstract

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.

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## Acknowledgements

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).

## Author information

Authors

### Contributions

M.H., F.L., O.S., U.H., S.W.K., M. Kira and R.H. conceived the study. M.H., F.L., O.S., M. Knorr and R.H. carried out the experiment and analysed the data. U.H., S.W.K. and M. Kira developed the quantum-mechanical model and carried out the computations. M.H., F.L., O.S., U.H., S.W.K., M. Kira and R.H. wrote the manuscript. All authors discussed the results.

### Corresponding authors

Correspondence to M. Kira or R. Huber.

## Ethics declarations

### Competing interests

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.

### Extended Data Figure 2 Determination of the absolute timescale.

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.

### Extended Data Figure 3 Double-blind reconstruction of the experimental XFROG data.

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).

### Extended Data Figure 4 Microscopic model.

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 $d h 1 e 1$, 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.

### Extended Data Figure 5 Quantum interference paths.

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 .

### Extended Data Figure 6 Non-perturbative quantum interference.

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).

### Extended Data Figure 7 Temporal phase of the HH emission from GaSe.

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.

### Extended Data Figure 8 HH pulse shaping via the CEP of the driving waveform.

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).

### Extended Data Figure 9 Ultrashort pulse shaping via spectral amplitude filtering.

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).

### Extended Data Figure 10 Triangle system and symmetry.

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.

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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

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