Abstract
Solids exposed to intense electric fields release electrons through tunnelling. This fundamental quantum process lies at the heart of various applications, ranging from high brightness electron sources in d.c. operation1,2 to petahertz vacuum electronics in laser-driven operation3,4,5,6,7,8. In the latter process, the electron wavepacket undergoes semiclassical dynamics9,10 in the strong oscillating laser field, similar to strong-field and attosecond physics in the gas phase11,12. There, the subcycle electron dynamics has been determined with a stunning precision of tens of attoseconds13,14,15, but at solids the quantum dynamics including the emission time window has so far not been measured. Here we show that two-colour modulation spectroscopy of backscattering electrons16 uncovers the suboptical-cycle strong-field emission dynamics from nanostructures, with attosecond precision. In our experiment, photoelectron spectra of electrons emitted from a sharp metallic tip are measured as function of the relative phase between the two colours. Projecting the solution of the time-dependent Schrödinger equation onto classical trajectories relates phase-dependent signatures in the spectra to the emission dynamics and yields an emission duration of 710 ± 30 attoseconds by matching the quantum model to the experiment. Our results open the door to the quantitative timing and precise active control of strong-field photoemission from solid state and other systems and have direct ramifications for diverse fields such as ultrafast electron sources17, quantum degeneracy studies and sub-Poissonian electron beams18,19,20,21, nanoplasmonics22 and petahertz electronics23.
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Data availability
An example of the resonance extraction and removal code is provided and can be directly download from https://github.com/philipLPFAU/Rate_extraction_Nature.git or accessed through https://doi.org/10.5281/zenodo.7612029. For convenience, we summarize the required TDSE model of ref. 27 in the Supplementary Information. Source data are provided with this paper.
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Acknowledgements
This work has been supported in part by the European Research Council (Consolidator Grant NearFieldAtto and Advanced Grant AccelOnChip) and the Deutsche Forschungsgemeinschaft (priority program no. SPP 1840 QUTIF). T.F. and L.S. acknowledges financial support from the Deutsche Forschungsgemeinschaft within the Heisenberg programme (grant nos. 315210756, 398382624 and 436382461) and by CRC 1477 ‘Light-Matter Interactions at Interfaces’ (grant no. ID 441234705). A. Liehl and A. Leitenstorfer acknowledge funding by DFG, grant no. 425217212–SFB 1432. We thank N. Dudovich, M. Krüger and C. Lemell for discussions.
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P.D. and T.P. measured the photoelectron spectra. A. Liehl and A. Leitenstorfer designed and fabricated the pulse compression fibre. P.D., T.P. and L.S. analysed the data and generated the plots. L.S., P.D. and T.F. performed the numerical simulations. P.D., L.S., T.F. and P.H. wrote the manuscript with input from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Different evaluation methods for the optimal phase.
For each energy E we fit the rate with a cosine function and define the optimal phase as the phase maximizing the rate (shown here for the measurement). The semi-transparent orange band indicates the 1σ confidence interval of the fits with average error of ± 0.04π. The considered relative phase interval extends from −2π to 3π. Alternatively, the optimal phase can be extracted by tracking the FFT-phase (black curve) or the maximum count rate (blue lines). In the latter method the resolution is restricted to discrete relative phase steps of 0.1π.
Extended Data Fig. 2 Impact of the field admixture.
a, Relative phases \({\phi }_{{E}_{{\rm{m}}{\rm{a}}{\rm{x}}}}\) maximizing the cut-off energy and b, corresponding peak-to-peak cut-off modulation depths Epp as function of field admixture α (lower horizontal axis) or the respective intensity admixture (upper axis) extracted from TDSE simulations (blue symbols) and the experiment (orange symbols). Solid lines indicate the predictions of the simple man’s model (SMM). The dashed line is a linear fit of the TDSE results. c, Phase-averaged photoelectron spectra of experiment (orange) and TDSE as blue shaded band containing all field admixtures from 5 to 27%. d, Optimal phase for the same field admixture range and with the same colour code as in panel c. Clearly, the overall shape of the spectra (c) and optimal phase (d) hardly vary when the relative field strength of the second harmonic is varied in the large range of 5 to 27% of the fundamental field. Most importantly, this highlights the robustness of TCMS for extracting the attosecond emission time window via the optimal phase independent of the field admixture.
Extended Data Fig. 3 Relation between optimal phase and work function.
Optimal phases (a,d), birth time and relative phase-dependent instantaneous emission rates (b,e) and phase-dependent yields (c,f) as in Fig. 3c–e, extracted from the TDSE simulation for work functions W = 4 eV (a–c) and W = 8 eV (d–f). Small/large work functions result in temporal broadening/confinement of the emission window (cf. Fig. 3f), which (following the argumentation in Fig. 3) shifts the optimal phase in the cut-off domain away from/towards ϕrel = π. This is most clearly visible in the right-most column, where the spread of the optical phases for the three spectral regions is much larger for W = 4 eV (top row) than for W = 8 eV (bottom row). This fully explains the behaviour of the optimal phase and, reversely, allows us to extract the electron emission duration with high precision.
Extended Data Fig. 4 Experimental setup and tip characterization.
a, Experimental setup consisting of a pulse shortening stage, second harmonic generation (SHG) stage, dichroic Mach-Zehnder interferometer and UHV-vessel containing the needle tip and the electron spectrometer. Abbreviations: half-mirror (D-M), prism compressor (PC), fibre collimator (FC), single-mode fibre (SMF), highly non-linear fibre (HNLF), 90° off-axis parabola (OAP), bismuth borate crystal (BIBO), fused silica plate (SiO2), dichroic beam splitter (DBS), half-wave plate (λ/2), polarization filter (PF), neutral density filter (ND) and chirped mirror (CM). Top right insets: pulse shapes, pulse durations and percentages of energy contained in the main peaks determined from frequency resolved optical gating (FROG) measurements of the fundamental (red) and second harmonic pulses (blue). b, Field ion microscopy image of the tip in preparation chamber. The ring counting method shows ~8 rings from the (110) to (211) pole corresponding to an apex radius of ~15 nm. c, Field ion microscopy image in experimental chamber. d, Field emission image for bias voltage of U = −390 V and e, laser emitted electrons for bias voltage of U = −100 V.
Extended Data Fig. 5 Cut-off extraction and relation to the ponderomotive energy.
a, Single-colour TDSE spectra for the fundamental field with increasing ponderomotive energy Up. b, Cut-off energies evaluated from single-colour TDSE simulations (blue circles) and experimental electron spectra (orange and black circles) as function of the ponderomotive energy Up. The cut-off position for the fundamental field strength used in the two-colour experiment is highlighted in orange. Both the cut-off positions from the simulated and the experimental data points depend linearly on the ponderomotive energy, but slightly deviate from the 10Up and 10Up+0.5W rules (dashed lines).
Extended Data Fig. 6 Trajectory Analysis.
a, Final kinetic energies (black curves) in units of the ponderomotive energy within the central cycles of a two-colour field (for field and vector potential see top panel) obtained from SMM calculations at a field admixture of 20% (phases as indicated). b,c, Solid curves show the CEP-averaged peak-to-peak cut-off modulation depths Epp (b) and relative phases \({\phi }_{{E}_{{\rm{m}}{\rm{a}}{\rm{x}}}}\) (c) resulting in maximal energies as function of field admixture as predicted by SMM. Dashed curves indicate respective results when neglecting the modification of the rescattering times due to the presence of the second harmonic (i.e. α and ϕrel). Shaded areas indicate the small variations of the respective properties with the CEP.
Extended Data Fig. 7 Time-dependent wavefunction and continuum resonances.
a, Probability density of the propagated wavefunction Ψ in logarithmic colour-scale with ground state indicated on the right (blue). b, Probability density after projecting out the ground state. c, Binding potential perturbed by electric field E (gray). The unperturbed ground state Ψground and first four continuum resonances \({\varPsi }_{{\rm{r}}{\rm{e}}{\rm{s}}}^{(1)}\) to \({\varPsi }_{{\rm{r}}{\rm{e}}{\rm{s}}}^{(4)}\) associated with the perturbed potential are indicated.
Extended Data Fig. 8 Successive removal of continuum resonances and extraction of the instantaneous emission rate.
a, Probability density after projecting out the ground state (same as Extended Data Fig. 7b). b-d, Remaining density after successively removing the respective resonances (shown on the right). e, Yield (orange) and instantaneous rate (blue) evaluated.
Extended Data Fig. 9 Comparison between full and truncated-field time propagation.
a,b, Probability density of the wavefunction after removing \({\varPsi }_{{\rm{r}}{\rm{e}}{\rm{s}}}^{(0)}\) to \({\varPsi }_{{\rm{r}}{\rm{e}}{\rm{s}}}^{(3)}\) including (a) and without (b) rescattering by truncating the field for positions x > 10 Å (dashed line). c, Yield (orange) and rate (blue) determined from (a) (dashed lines) and (b) (solid lines).
Extended Data Fig. 10 Least square optimization of work function and mapping to emission duration.
a, The minimum in the least square optimization is found at W = 6.61 eV (blue curve) and matches the experimental optimal phase (orange) in the cut-off domain. The 1σ and 3σ confidence intervals are indicated for σ = 0.32 eV. b, Emission duration (FWHM) depending on work function for ϕce = π. Blue circles indicate CEP-averaged durations for W = 6.3 eV, 6.6 eV and 6.9 eV. The grey area indicates the previously determined work function interval. The maximum variation of the emission duration within this interval of 710 ± 30 as is highlighted by the blue shaded area.
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Dienstbier, P., Seiffert, L., Paschen, T. et al. Tracing attosecond electron emission from a nanometric metal tip. Nature 616, 702–706 (2023). https://doi.org/10.1038/s41586-023-05839-6
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DOI: https://doi.org/10.1038/s41586-023-05839-6
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