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Few-electron correlations after ultrafast photoemission from nanometric needle tips

Nature Physics volume 19, pages 1402–1409 (2023)Cite this article

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Abstract

Free electrons are essential in such diverse applications as electron microscopes, accelerators and photoemission spectroscopy. However, the space charge effects of many electrons are often a problem and, when confined to extremely small space–time dimensions, even two electrons can interact strongly. Here we demonstrate that the resulting Coulomb repulsion can be highly advantageous, as it leads to strong electron–electron correlations. We show that femtosecond laser-emitted electrons from nanometric needle tips are highly anticorrelated in terms of energy because of dynamic Coulomb repulsion, with a visibility of 56%. We extract a mean energy splitting of 3.3 eV and a correlation decay time of 82 fs. The energy-filtered electrons display a sub-Poissonian number distribution with a second-order correlation function as small as g(2) = 0.34, implying that shot-noise-reduced pulsed electron beams can be realized by simple energy filtering. We also reach the strong-field regime of laser-driven electron emission and gain insights into how the electron correlations of the different electron classes (direct or rescattered) are influenced by the strong laser fields.

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Fig. 1: Schematic of the experiment.
Fig. 2: Energy correlation in two-electron events.
Fig. 3: Second-order correlation maps and electron number distributions.
Fig. 4: Ultrafast temporal behaviour of the anticorrelations from a two-pulse measurement with adjustable time delay.
Fig. 5: Laser intensity dependence of the observed anticorrelation gap and strong-field effects.
Fig. 6: Energy gap size from quantum and semi-classical 1D simulations.

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Data availability

Source data are provided with this paper. All other data that support the plots within this Article and other findings of this study are available from the corresponding author upon reasonable request.

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Acknowledgements

We thank A. Czasch for technical discussions on the delay-line detector and P. Dienstbier for discussions on the semi-classical simulation. This research was supported by the European Research Council (Consolidator Grant NearFieldAtto and Advanced Grant AccelOnChip) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project-ID 429529648 - TRR 306 QuCoLiMa (‘Quantum Cooperativity of Light and Matter’) and Sonderforschungsbereich 953 (‘Synthetic Carbon Allotropes’), Project-ID 182849149. J.H. acknowledges funding from the Max Planck School of Photonics.

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Author notes

  1. These authors contributed equally: Stefan Meier, Jonas Heimerl.

Authors and Affiliations

  1. Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany

    Stefan Meier, Jonas Heimerl & Peter Hommelhoff

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  1. Stefan Meier
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Contributions

S.M. and J.H. performed the experiment, analysed the data and generated the plots. S.M. performed the semi-classical simulations and J.H. performed the quantum-mechanical simulations. All authors wrote the manuscript.

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Correspondence to Stefan Meier.

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Nature Physics thanks Kasra Amini, Phillip Keathley and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 Measurement scheme for heralded of electrons.

Electron events triggered from a metal needle tip are energetically separated by an Omega filter. When the energy width of single electron events is smaller than the mean Coulomb energy splitting, the electrons can be separated after the omega filter. Single and two-electron events are then separated in absolute energy. The measurement of one electron in the energy region prohibited for single electron events, only possible for two-electron events, leads directly to the knowledge of the presence of a second electron. By post-selection a deterministic electron source is achieved, which can be used for quantum imaging.

Extended Data Fig. 2 Sketch of experimental setup.

Femtosecond laser pulses focused by an off-axis parabolic mirror (OAP) trigger electrons (blue) from a metal needle tip. The highly coherent electron beam is magnified by two quadrupoles by a factor of up to 10 (only one quadrupole shown here). The two quadrupoles can be moved by a 3-axis manipulation stage. The multi-hit capability of the delay-line detector allows us to measure the position x and y, and the time of flight for each electron individually.

Extended Data Fig. 3 Power scaling of n-electron events.

Shown is the laser power vs. the counts per laser shot on a double-logarithmic scale. Because of the multi-photon photoemission process, the electron emission probability P follows a power law \(P\propto {I}_{L}^{{n}_{{{{\bf{ph}}}}}}\), visible by the linear scaling in the double-logarithmic representation. Here, nph denotes the number of absorbed photons. The slopes are m = 2.8 ± 0.4 (one electron, red), m = 6.5 ± 0.5 (two electrons, blue) and 8.9 ± 0.7 (three electrons, green). The slope for the total emission (sum of all events) is m = 3.3 ± 0.4. The linear fits were weighted by the total count rate for each laser power. The inset shows the corresponding slopes for the one, two and three electron slopes with a linear fit (black line).

Source data

Extended Data Fig. 4 Second order correlation function \({g}^{(2)}({E}_{1},{E}_{2})=\frac{ < I({E}_{1})I({E}_{2}) > }{ < I({E}_{1}) > < I({E}_{2}) > }\) calculated for the energy map shown in Fig. 2(a).

A filter excluding data points with less than 5 counts per bin removes data points dominated by noise (gray mask).

Extended Data Fig. 5 Electron energy filtering to shape the emission statistics.

a) Explanation of the energy filter used in Fig. 3: a central energy is chosen (orange line) and then an interval of ± Ewindow/2 is used as filter area. (b) Experimental data shown in Fig. 3(a), with highlighted variables. (c) Fano factor for simulations from an average of 0.1 electron per pulse to 5 electrons per pulse being emitted. The insets show the resulting energy-filtered hit-distributions after propagation including Coulomb interactions.

Source data

Source Data Fig. 2

Measurement data.

Source Data Fig. 3

Measurement data.

Source Data Fig. 4

Measurement data.

Source Data Fig. 5

Measurement data.

Source Data Fig. 6

Simulation data.

Source Data Extended Data Fig. 3

Measurement data.

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Meier, S., Heimerl, J. & Hommelhoff, P. Few-electron correlations after ultrafast photoemission from nanometric needle tips. Nat. Phys. 19, 1402–1409 (2023). https://doi.org/10.1038/s41567-023-02059-7

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