Abstract
Strong light fields have created opportunities to tailor novel functionalities of solids1,2,3,4,5. Floquet–Bloch states can form under periodic driving of electrons and enable exotic quantum phases6,7,8,9,10,11,12,13,14,15. On subcycle timescales, lightwaves can simultaneously drive intraband currents16,17,18,19,20,21,22,23,24,25,26,27,28,29 and interband transitions18,19,30,31, which enable high-harmonic generation16,18,19,21,22,25,28,29,30 and pave the way towards ultrafast electronics. Yet, the interplay of intraband and interband excitations and their relation to Floquet physics have been key open questions as dynamical aspects of Floquet states have remained elusive. Here we provide this link by visualizing the ultrafast build-up of Floquet–Bloch bands with time-resolved and angle-resolved photoemission spectroscopy. We drive surface states on a topological insulator32,33 with mid-infrared fields—strong enough for high-harmonic generation—and directly monitor the transient band structure with subcycle time resolution. Starting with strong intraband currents, we observe how Floquet sidebands emerge within a single optical cycle; intraband acceleration simultaneously proceeds in multiple sidebands until high-energy electrons scatter into bulk states and dissipation destroys the Floquet bands. Quantum non-equilibrium calculations explain the simultaneous occurrence of Floquet states with intraband and interband dynamics. Our joint experiment and theory study provides a direct time-domain view of Floquet physics and explores the fundamental frontiers of ultrafast band-structure engineering.
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Data availability
The data supporting the findings of this study are available at http://doi.org/10.5283/epub.53924.
Code availability
Program codes for reproducing the theory calculations are available at http://doi.org/10.5283/epub.53924.
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Acknowledgements
We thank K. Richter and Z. Tao for discussions. The work in Marburg has been supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through Project ID 223848855-SFB 1083 and Research Grants GU 495/2 and HO 2295/9. M.S. thanks the Swiss National Science Foundation SNF for its support with an Ambizione grant (project number 193527). The work in Regensburg has been supported by the Deutsche Forschungsgemeinschaft (DFG) through Project ID 422 314695032-SFB 1277 (Subproject A05) as well as Research Grant HU1598/8. O.E.T. and K.A.K. have been supported by the RFBR and DFG (project number 21-52-12024) in the part of crystal growth and the Russian Science Foundation (project number 22-12-20024) in the part of crystal characterization and state contract of IGM SB RAS and ISP SB RAS. S.I. acknowledges support from JSPS Postdoctoral Fellowship for Research Abroad. M.A.S. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the Emmy Noether Programme (SE 2558/2).
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S.I., J.G., U.H. and R.H. conceived the experiment. S.I., M.M., S.S., J.F., J.R., D.A. and J.G. carried out the experiment. K.A.K. and O.E.T. grew the crystals and characterized their properties. M.S. and M.A.S. developed and carried out the quantum non-equilibrium calculations. S.I. carried out the DFT calculations and developed the minimal quantum model. All authors analysed the data and discussed the results. S.I., U.H. and R.H. wrote the paper with contributions from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Lightwave ARPES set-up.
a, The output of a titanium-sapphire (Ti:Sa) amplifier system is used to generate intense MIR pulses with stable CEP by DFG of the near-infrared pulse trains from a dual-branch OPA (left half). In parallel, broadband UV probe pulses are generated (right half), which are spatially overlapped with the MIR field transients and guided into the UHV chamber for ARPES. Arrows indicate the polarization state of each beam. b, MIR waveform measured by electro-optic sampling. Inset: corresponding amplitude spectrum. c, Spectral intensity of the laser fundamental (black) and the MPC (red). d, Broadband UV spectrum (FWHM, 25 nm). e, Measured (top) and reconstructed (bottom) XFROG traces of the UV probe pulse. f, Intensity and phase profiles of the reconstructed UV pulse providing a duration of 17 fs (FWHM). g, Dependence of the 2PPE spectra on the position of prism P4. The maximum 2PPE intensity is reached when the UV pulses in the UHV chamber are optimally compressed.
Extended Data Fig. 2 Validation of broadband 2PPE probing and curvature image processing.
a, One-photon ARPES map of Bi2Te3 measured with 200-nm probe light with a bandwidth of 2 nm. Grey curves and shaded areas depict the DFT band structure of Bi2Te3. b, 2PPE-ARPES map of Bi2Te3 measured with the broadband 400-nm probe. c,d, Curvature image filtering of the 2PPE spectrum of b precisely retrieves the band structure obtained with one-photon ARPES in a. The band visibility is controlled by the factor C0 in Eq. (1) and the level of the preceding smoothing process. Panels c and d compare 2D curvature-filtered results obtained with a preliminary smoothing filter size of 0.035 Å−1 × 0.06 eV and 0.05 Å−1 × 0.09 eV, respectively.
Extended Data Fig. 3 Reconstruction of electric-field and current waveforms.
a, Momentum-streaking trace obtained by integrating raw ARPES spectra in the region of the bulk valence band. b, Curvature-filtered momentum-streaking trace of the same data. c, Momentum-streaking waveform extracted from b (blue) and resulting s-polarized electric-field waveform (black). Open circles indicate data points, the solid line is a fit to the analytical waveform, see Eq. (2). Inset: amplitude spectra of the Fourier-transformed field waveform. d, Curvature-filtered lightwave ARPES maps measured at t = −100 fs, 0 fs, and 20 fs (vertical lines in e). Blue and red boxes indicate areas where the electron occupation is integrated for the determination of surface currents. e, Upper panel: electric field waveform from (c). Lower panel: current-density waveform extracted from ARPES data (red circles) and calculated from the measured electric field by means of the semiclassical Boltzmann equation (red line).
Extended Data Fig. 4 THz field-strength dependence of Floquet sidebands.
a, Measured MIR driving field with a centre frequency of 25 THz and a peak field strength of 0.5 MV cm−1. Circles, experimental data from streaking reconstruction, solid line: numerical fit of Eq. (2). b, Curvature-filtered lightwave ARPES maps at t = −100 fs, −54 fs, −46 fs, and −38 fs. These temporal positions are indicated in a. c,d, Comparable data set for a peak field strength of 0.8 MV cm−1 (data set shown in Fig. 2).
Extended Data Fig. 5 Sideband signatures in raw ARPES maps.
a, Left, the curvature-filtered lightwave ARPES map at t = −40 fs adapted from Fig. 2. Right, the corresponding raw ARPES image. The solid and the dashed boxes depict the regions considered to extract the energy distribution curves in b and the one-dimensional (1D) intensity distribution curves in c, respectively. b, An energy distribution curve obtained from the raw ARPES map (orange solid line) and its fit (black solid line) by using two Gaussian peaks (grey shaded areas) and an exponential background (black dotted line). The orange dashed line shows an energy distribution curve extracted from the curvature image, and the vertical grey lines highlight energy splitting, which is quantitatively consistent with the driving frequency of 25 THz. c, 1D intensity distribution curves in the direction perpendicular to the left Dirac branch (indicated by AB in a) from raw ARPES data measured at t = −60 fs, −50 fs, −48 fs, −40 fs, and −34 fs. The curves are normalized and vertically offset for clarity. The orange lines show raw intensity curves averaged inside the extraction region, and the black lines are smoothed by a 1D filter. Blue arrows highlight positions of emergent peaks in the raw data, and the dashed orange line is a guide tracking the evolution of the ground-state peak.
Extended Data Fig. 6 Systematic comparison of curvature-filtered and raw ARPES spectra.
a, The curvature-filtered lightwave ARPES maps measured at t = −60 fs, −50 fs, −40 fs, −18 fs, −12 fs, and 120 fs adapted from Figs. 2 and 4. b, The corresponding raw ARPES images. Overlaid white lines depict 2D contours of the ARPES intensities. Formation of additional peaks in regions where there is no equilibrium band dispersion is clearly visible in the raw spectra. In addition, strong suppression of the bulk intensity, which spreads into Floquet–Bloch bands, is directly observed in the intensity distributions especially at t = −18 fs and t = −12 fs. After the driving field leaves the surface at t = 120 fs, the system basically recovers the original spectral feature, except for the persistent intensities in the bulk conduction band and the enhanced occupation in the surface states due to finite heating.
Extended Data Fig. 7 Observation of Floquet–Volkov sidebands with mixed s- and p-polarized electric fields.
a, s-polarized and b, p-polarized MIR electric field waveform (centre frequency, 25 THz) reconstructed from momentum and energy streaking of a single lightwave ARPES measurement, respectively. c, Streaking-compensated, curvature-filtered lightwave ARPES maps measured at t = −90 fs, −8 fs, −4 fs, 32 fs, and 90 fs (vertical lines in a and b) with the driving field characterized in a and b, featuring both s- and p-polarization components.
Extended Data Fig. 8 Absence of resonance effects in 2PPE with broadband 400-nm probe pulses.
a, Spectra of a picosecond, wavelength-tunable light source (blue curves) and the broadband UV probe used for lightwave ARPES (orange). The intensity of the blue spectra was adjusted for constant photon flux. b, 2PPE-ARPES maps measured with the tunable probe for different wavelengths (photon energies) as indicated in the respective panels. c, Integrated 2PPE intensity, I2PPE, as a function of the probe centre wavelength. The solid curve is a guide to the eye.
Extended Data Fig. 9 Model band structure of Bi2Te3 and time-resolved ARPES simulations.
a, Band structure used in our lightwave ARPES calculations. Left, the pure TSS band structure given by the \({\bf{k}}\cdot {\bf{p}}\) Hamiltonian (Eq. 7). Right, the 3B model used to describe dynamical Floquet–bulk coupling. The grey shaded area illustrates delocalization of the bulk states to account for self-energy effects in our model (see Methods). b, Input driving waveform which well reproduces the experimental result in Fig. 2b (black curve) and sech-shaped probe pulses with a bandwidth of 47 meV (intensity FWHM) (filled blue curve). Thin vertical lines indicate the temporal positions shown in c. c, Curvature-filtered ARPES images for the 47 meV probe calculated at different time steps with the pure TSS band structure (red curve in a, see also Figs. 2 and 4 for corresponding experimental data). d, Corresponding raw ARPES spectra. e, ARPES spectra for different bandwidths and shapes of the probe pulse at a fixed delay time t = −40 fs.
Extended Data Fig. 10 Systematic lightwave ARPES at MIR frequencies.
a, MIR waveform centred at 29 THz. Circles represent the reconstruction from momentum streaking, the solid curve is a numerical fit of Eq. (2). b, Curvature-filtered lightwave ARPES maps measured before the arrival of the THz field (t = −120 fs), at three temporal positions during the field transient (vertical lines in a), and after the THz field has left the surface (t = 120 fs). c,d, Corresponding data set obtained for a centre frequency of 31 THz. e,f, Corresponding data set for a centre frequency of 41 THz. The larger splitting at 41 THz (photon energy, 0.16 eV) facilitates the coupling between Floquet sidebands and the bulk conduction band already at lower field strengths than in the case of excitation at 25 THz (Fig. 4).
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Ito, S., Schüler, M., Meierhofer, M. et al. Build-up and dephasing of Floquet–Bloch bands on subcycle timescales. Nature 616, 696–701 (2023). https://doi.org/10.1038/s41586-023-05850-x
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DOI: https://doi.org/10.1038/s41586-023-05850-x
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