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High-harmonic spectroscopy probes lattice dynamics

Abstract

The probing of coherent lattice vibrations in solids has conventionally been carried out using time-resolved transient optical spectroscopy, with which only the relative oscillation amplitude can be obtained. Using time-resolved X-ray techniques, absolute electron–phonon coupling strength could be extracted. However, the complexity of such an experiment renders it impossible to be carried out in conventional laboratories. Here we demonstrate that the electron–phonon, anharmonic phonon–phonon coupling and their relaxation dynamics can be probed in real time using high-harmonic spectroscopy. Our technique is background-free and has extreme sensitivity directly in the energy domain. In combination with the optical deformation potential calculated from density functional perturbation theory and the absolute energy modulation depth, our measurement reveals the maximum displacement of neighbouring oxygen atoms in α-quartz crystal to tens of picometres in real space. By employing a straightforward and robust time-windowed Gabor analysis for the phonon-modulated high-harmonic spectrum, we successfully observe channel-resolved four-phonon scattering processes in such highly nonlinear interactions. Our work opens a new realm for the accurate measurement of coherent phonons and their scattering dynamics, which allows for potential benchmarking ab initio calculations in solids.

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Fig. 1: The HHS-based principle of measuring lattice vibration dynamics.
Fig. 2: Experimental and simulated time-delayed THG trace of lattice dynamics from HHS.
Fig. 3: Optical manipulation of the electronic and phononic properties of α-quartz crystal.
Fig. 4: Reconstructed oxygen-atom displacement dynamics of the two optical modes A1g and A1b by combining the calculated ODP from DFPT.
Fig. 5: Phonon dispersion and possible anharmonic decays for A1g and A1b phonons by a four-phonon scattering pathway.

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All data and codes in the main text or the Extended data materials are available from the authors upon reasonable request.

Code availability

All data and codes in the main text or Extended data materials are available from the authors upon reasonable request.

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Acknowledgements

We thank M. Molitor for discussions. We acknowledge funding support in part from the ETH Zurich Postdoctoral Fellowship Program (FEL–31 15–2), the Marie Curie Actions for People COFUND Program and SNSF R’equip grant no. 206021_170775, and in part from the Department of Physics, Faculty of Science (HKU), RGC ECS project 27300820, GRF project 17315722 and Area of Excellence project AoE/P-701/20. D.W. is funded by Science Foundation Ireland (SFI) under grant no. 18/RP/6236.

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Authors and Affiliations

Authors

Contributions

J.Z. and Z.W. constructed the experimental set-up. J.Z. and T.T.L. collected the data. J.Z. performed the analysis and simulations based on the two-level quantum model proposed by F.L., D.W., D.E.R. and T.K. J.Z. and T.T.L. performed the data fitting and time–frequency analysis. Z.W. and J.Z. conducted the electronic and phononic related calculations from first principles. All authors contributed to the writing of the manuscript, which was first drafted by J.Z. H.J.W. and T.T.L. supervised the project. All authors discussed and interpreted the experimental data.

Corresponding authors

Correspondence to Doris E. Reiter, Tilmann Kuhn, Hans Jakob Wörner or Tran Trung Luu.

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Extended data

Extended Data Fig. 1 Electronic bandstructure of quartz crystal.

a, Unit cell of bulk α-quartz crystal which consists of nine atoms (three Si and six O atoms) in total and b, the corresponding Brillouin zone. c and d, Calculated electronic bandstructure and density of states from DFT.

Extended Data Fig. 2 Energy-calibrated, spatially resolved HHG spectra.

a, 400 nm only. b,800 nm only. c, Combination of them after vertical integration on the MCP image.

Extended Data Fig. 3 Measured and Simulated Time-resolved HHG spectra of 400 nm pulses.

a and c, Time-resolved spectrum trace of 4th and 5th harmonic of 400 nm probe. b, and d, The corresponding simulations from the quantum model. The color bar is linear and normalized to the maximal value of the harmonic signal.

Extended Data Fig. 4 Phonon mode and phonon dispersion curve of quartz crystal.

a and b, Vibrational modes of A1g and A1b phonon modes of α-quartz in the z-direction of the lattice, the arrows and their lengths indicate the directions of atomic motion and oscillation amplitudes. c and d, Phonon dispersion and phonon density of states that were calculated from DFPT.

Extended Data Fig. 5 Bandgap variation at different probe pulse field strengths.

a and b, Denote the values extracted from experimental spectra that were measured in Γ-M and Γ-K direction when the pump intensity was fixed aroung 1.1 V/Å, respectively. The error bars for the bandgap correspond to the standard deviation of the intensity over 5 independent measurements.

Extended Data Fig. 6 Intensity scalling of THG of 400 nm probe.

The blue dot line represents the measured THG yields at varying laser intensities, while the red solid line corresponds to the fitting results obtained using a perturbative power law model In (I is the pulse intensity and n is the harmonic order, here n = 3). The yellow line is a linear fit to the experimental data. The error bars for THG yields correspond to the standard deviation of the intensity over 3 independent measurements, with each measurement in 4 million laser pulses.

Extended Data Fig. 7 Quantifying ph-ph couplings from the classical coupled oscillator model.

a and b, Modulated spectrum of A1g and A1b modes with probe intensity-scaling measurement when both pump and probe pulses are polarized in Γ−M direction. The legends show the estimated probe peak intensity ranging from 0.161 to 0.170 V/Å, and fixed pump field strength ~1.0 V/Å. The error bars for each quantity (a-b) correspond to the standard deviation of the intensity over 5 independent measurements. The solid line indicates the linear fitting according to the coupled classical oscillator interaction model. c, Extracted modulation frequency variations for both modes, the dashed lines represent the experimentally observed E128 and E262 phonon modes15. d, Anharmonic dimensionless ph-ph interaction constants. e and f, Initial phases of the modulation frequencies and corresponding relative delay time of connected phonon modes. The error bars in c to f include fitting, statistical and systematic uncertainties. The error bars for each quantity (c-f) correspond to the standard deviation of the intensity over 5 independent measurements, with each measurement in 2 million laser pulses.

Extended Data Fig. 8 Time-frequency and FFT analysis of the modulated spectra.

a. Time-frequency analysis of the modulated spectra from Extended Data Fig. 8a, b in the Γ−M direction. The solid green lines indicate the spacing frequencies of the specific phonon modes that contribute to the four-phonon scattering process. The intensity of all traces is normalized to their maximum. c and d, direct FFT of the modulated spectra in Fig. 8a and 7b, respectively. The arrows indicate the main peaks of the spectra.

Extended Data Fig. 9 Time-Frequency analysis (GT) under different time window widths.

a, b, and c. The GT Spectrum variation under different time window widths from 150 f, 100 fs, and 50 fs, respectively.

Extended Data Fig. 10 Phonon parameters fitting and residual errors.

a. Measured (dot) and fitted (solid line) COM spectrum of THG (upper panel), and the lower panel represents the residual error between the fitted and the measured data. b. Iterated SSE value variations after setting four pre-values off by 50 %.

Supplementary information

Supplementary information

Supplementary Table 1 | Comparison of the phonon frequencies. Energy (A and E symmetry) at the Γ point based on DFPT calculations in this work and the previous experimental Raman spectroscopy measurements. Supplementary Table 2 | Fitted phonon parameters of A1b and A1g modes. The fitted phonon parameters of Extended Data Fig. 10 are based on equation (3) in the main text.

Supplementary Video 1

Visualizing the lattice vibrations with HHS technique.

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Zhang, J., Wang, Z., Lengers, F. et al. High-harmonic spectroscopy probes lattice dynamics. Nat. Photon. (2024). https://doi.org/10.1038/s41566-024-01457-4

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