Main

Correlations between electrons are at the core of numerous phenomena in atomic, molecular and solid-state physics. Mediated by the Coulomb force, few- and many-body electronic correlations govern intriguing phases of matter, such as superconductivity or charge ordering, and they underpin a wide variety of applications down to nanoscale single-electron sources1,2 and logic gates based on single charges3,4. In contrast to the opportunities offered by electron correlations in condensed matter, Coulomb interactions in free-electron beams are usually considered to have adverse effects. In electron microscopy, electron repulsion leads to stochastic longitudinal and transverse emittance growth of the beam, described by the Boersch5 and Loeffler6 effects, respectively, and limiting the brightness of state-of-the-art electron sources7,8. In high-charge electron pulses for time-resolved experiments, mean-field and stochastic Coulomb effects govern the achievable pulse duration, energy spread and focusability, and pose a major experimental challenge for ultrafast electron diffraction9,10,11,12,13,14 and microscopy15,16,17, particle accelerators18 and free-electron lasers19.

Studying strong electronic correlations in a beam containing only a few particles requires the preparation of a sufficient electron phase space degeneracy. Field emitters represent highly localized sources, and they have been used in studies elucidating free-electron correlations20,21,22. In particular, the physical origin of antibunching in free-electron beams, as reported by Kiesel et al.20, has been a long-standing question and is still actively discussed in the context of exchange-mediated21 and Coulomb23,24 interactions. Tailoring such correlations in free-electron beams facilitates sub-Poissonian beam statistics22, promising shot-noise-reduced imaging and lithography. Strong interparticle interactions are enabled by spatiotemporally confined femtosecond-pulsed photoemission from nanotips25,26,27,28,29,30,31,32, employed for ultrafast electron microscopy and diffraction with high-coherence beams13,16,33. The pulse-averaged effects of Coulomb interactions from such sources have recently been investigated, and are associated with spectral broadening and a loss of temporal and spatial resolution17,34,35,36.

Employing concepts from quantum optics37, correlations among free electrons have previously been identified by coincidence detection using detector pairs20,21,22,38, as in atomic and molecular science measuring electrons and ions39,40, correlated photoemission41,42,43 and ionization44,45.

In electron microscopy, the recent advent of pixelated event detectors has substantially broadened capabilities for coincidence measurements involving electrons, as demonstrated for electron-correlated X-ray emission46, cathodoluminescence at nanomaterials47 and integrated photonic resonators48. These capabilities will foster the emerging area of free-electron quantum optics, promising quantum coherent manipulation49,50,51,52,53,54 and sensing55,56 at the nanoscale, and facilitating concepts based on electron–electron20,57,58,59 or electron–light entanglement48,57,59,60,61,62. Establishing such schemes will require a fundamental and quantitative understanding of correlations between the single electrons in the beam.

Here, we demonstrate strong Coulomb correlations in few-electron states generated at a laser-driven Schottky field emitter. Using event-based electron spectroscopy and imaging, the kinetic energy distributions of electron ensembles emitted by single laser pulses are recorded, and events are sorted by the number of free electrons. Characteristic multi-lobed spectra for events containing two, three and four electrons are found. We quantitatively characterize interparticle correlations in terms of both energy and transverse momentum, and observe that these few-body interactions dominate over mean-field (space charge) effects. Two-electron energy correlation functions reveal pronounced peaks separated by energy differences of around 1.7 eV, illustrating an energy exchange facilitated by acceleration-enhanced longitudinal interaction along the beam axis. Transverse correlations in conjunction with transverse momentum selection cause antibunching and sub-Poissonian beam statistics. The relative contributions of longitudinal and transverse correlations, and thus the resulting antibunching factor, can be controlled by the initial acceleration field. The findings shed light on fundamental correlations in multi-electron pulses and enable statistical control of electron beams for on-demand correlated few-particle imaging and spectroscopy.

The experiments presented in this study were carried out at the Göttingen Ultrafast Transmission Electron Microscope (see sketch in Fig. 1a)16. Using a femtosecond laser source, ultrashort electron pulse trains at low pulse charge are generated by near-threshold laser-assisted Schottky emission from a W(100)/ZrO emitter. After propagation through the column of the microscope, the electrons are detected with an event-based camera. The temporal resolution of the electron detector allows consecutive incident electron pulses to be distinguished, providing an unambiguous measure of the number n of transmitted electrons for each laser pulse (Fig. 1a,b).

Fig. 1: Electron number states in event-based transmission electron microscopy.
figure 1

a, Experimental setup. Few-electron states are prepared by pulsed photoemission. The electrons pass the sample plane of the microscope and, via post selection, event-based electron spectroscopy enables number-state-selective beam analysis. b, Ultrashort electron pulses are emitted from a laser-assisted Schottky field emitter (W(100)/ZrO nanotip), with a pulse charge of up to a few electrons coupled to the microscope column. c, Power scaling of the rates of one-, two- and three-electron states with fitted slopes of 0.99 (n = 1), 1.99 (n = 2) and 2.95 (n = 3) on a double-logarithmic scale. d, Second-order correlation function g2(τ) of detected electrons with a timing resolution of approximately 10 ns. Inset: The peaks are spaced by the repetition time between laser pulses Trep. The strongly reduced correlation function at zero delay is a clear experimental signature of antibunching. e–i, The event-averaged spectrum (e) and separate number-state resolved contributions for n = 1 (f), n = 2 (g), n = 3 (h) and n = 4 (i). The two-, three- and four-electron spectra are magnified (see factors in panels) and show distinct shapes with n peaks, indicating discrete energetic separation.

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The rates of n-electron events as a function of incident laser power are displayed in Fig. 1c. Specifically, the rate of single-electron emission scales linearly with power, in agreement with the employed process of near-threshold laser-assisted Schottky photoemission16,63,64. Correspondingly, the n = 2 and n = 3 electron rates scale with to the power of n; that is, with the square and cube, respectively, of the laser power (only one measurement at high power was conducted for n = 4). Considering the relative distribution of n-electron events at a given laser power, we identify sub-Poissonian statistics. Specifically, defining P1 as the probability of detecting one electron in a pulse, a Poisson process predicts a probability of \({P}_{n}={P}_{1}^{n}/n!\) of detecting n electrons. The actual rates measured for n = 2 and n = 3 are lower, at only 85% and 57%, respectively, of those expected from the single-electron rate. This antibunching is also evident from a dip in the second-order correlation function at time delay τ = 0 (Fig. 1d), as discussed in detail below.

We next investigate the kinetic energies of these number-sorted electron states (Fig. 1e–i). The spectral distribution of the one-electron events (Fig. 1f), which also dominates the total spectrum (Fig. 1e, summed over all events), consists of a single peak centred on the acceleration energy of E0 = 200 keV. In stark contrast, the spectra of the few-electron events exhibit a number of lobes identical to the number of particles contained. In Fig. 1g–i, we plot the spectral distributions of the electron events sorted into event classes n = 2, 3, 4 with respect to the average energy \(\overline{E}\) of the electrons in each pulse. Extended Data Fig. 1 shows the spectra of the event classes with respect to the acceleration energy.

For the n = 2 events, this results in a histogram of energy differences—that is, the energy correlation function of the two-electron state. Depicted in terms of the magnitude of the energy difference in Fig. 2a, these measurements reveal a clear correlation gap of the energies of both electrons in a n = 2 state. A natural assumption would be that this energy gap arises from Coulomb interactions. A first question that needs to be answered is to what degree these correlations are modified by emitted electrons near the source that are not transmitted to the column, as such electrons are known to affect the overall spectral distribution7,8,15,16,17,65,66. We find that the correlation function is only weakly dependent on the laser power, and thus the average number of electrons (Extended Data Fig. 2). This shows that we are observing an effect that is governed primarily by the interaction of those few electrons within the measured ensemble.

Fig. 2: Coulomb-correlated few-electron pulses.
figure 2

a, The peak position of normalized one-sided pair correlation functions (n = 2) is nearly constant for varying laser power. b, Energy histogram of coincident n = 2 state electron pairs revealing a strong correlation in kinetic energy that is visible in the spectral correlation function (inset, integrated along the diagonal). c,d, The sorted energy histograms of n = 3 states (c) and n = 4 states (d) show clearly separated energy-pair correlation peaks of combinations Ei: EA−C and Ej: EB−D. e, Top: classical simulation scheme in a geometry consisting of emitter, extraction anode, second acceleration stage and aperture. Two electrons (blue dots) at the nanotip are injected into the static field with a temporal separation of Δt = 50 fs and are repelled by the inter-particle Coulomb force FC. Bottom: the momentary Coulomb energy EC (purple), electron velocity (green) and accumulated energy difference ΔE (magenta) are plotted against the electron travel distance from the emitter. A small initial Coulomb energy translates to a greatly enhanced final energy difference during acceleration. f, Final two-electron energy separations for varying emission time differences. The distributions of emission time differences for two Gaussian pulse shapes with a full-width-at-half-maximum of 200 fs at delays of 0 ps (green) and 0.4 ps (orange) are shown by the shaded regions. g, Pair correlation density of n = 2 states (colour scale) for photoemission with two delayed laser pulses. A strong correlation gap is observed at temporal overlap that gradually disappears for pulse delays >200 fs. The coloured arrows correspond to spectra in h. h, Comparison of pulse pair correlation spectra from g at temporal overlap (green) and a delay of 0.4 ps (orange) with simulated correlation spectra (dashed lines).

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The measurement scheme allows us to analyse the spectral characteristics in terms of two-dimensional energy correlation functions. Figure 2b shows the pair density distribution as a function of the electron energies EA and EB associated with two electrons A and B in the same electron pulse. The pairs exhibit a strong correlation gap around zero energy difference EA − EB. The broadening of the pair distribution in the average energy (EA + EB)/2 is found to depend more strongly on the laser power, illustrating that both electrons are affected jointly by an increase in stochastic interactions with electrons not entering the column (Methods). An analysis of the pair distribution of electron energies for the cases of n = 3 and n = 4 electrons (Fig. 2c,d) strikingly demonstrates a persistent, regular arrangement of the energies of electrons produced in a single pulse. These measurements highlight the pronounced interparticle Coulomb correlation at the level of 1–2 eV per electron, which we study further below.

To elucidate the physical origins of these strong correlations, we numerically simulate the particle propagation including the static acceleration field and interparticle Coulomb interactions. Specifically, we compute trajectories for sets of electrons with initial conditions representing the emission at the tip in terms of the distributions of initial momentum, emission location and temporal separation. Experimental parameters for the acceleration voltages and approximate electrode distances are used in the simulations, with further details provided in Supplementary Section C.

The most important findings of the simulations are a quantitative prediction and rationalization of the magnitude of the observed Coulomb correlations. Figure 2e illustrates the simulation result for an individual pair of electrons emitted with typical parameters for our experimental conditions. The electrons are extracted from the source with spatial and temporal separations of 8 nm and 50 fs, respectively. At the moment of emission of the second electron, the surface electric field of 0.5 V m−1 has already accelerated the first electron to a distance of 130 nm from the emitter surface, such that the initial transverse separation only accounts for a small fraction of the total particle distance. The electrostatic Coulomb energy at the time of emission of the second electron amounts to only 12 meV. Thus, the question of how such small electrostatic energies can translate to a final energy difference of 2 eV and higher arises.

We first provide a qualitative explanation of the enhanced correlation, using non-relativistic expressions for simplicity. In the absence of an accelerating field and for particles with mass m initially at rest, the Coulomb energy EC would only translate to a velocity difference of \(\Delta v={v}_{\mathrm{A}}-{v}_{\mathrm{B}}=2\sqrt{{E}_{\mathrm{C}}/m}\). However, considering the external acceleration of the particles to a mean velocity \(\bar{v}=({v}_{\mathrm{A}}+{v}_{\mathrm{B}})/2\), the same velocity difference results in a kinetic energy difference \(\propto \bar{v}\Delta v\) that is substantially larger than EC (see also ref. 67). Moreover, Coulomb energy is transferred to high kinetic energy differences only while the electrons are already at higher velocity in the laboratory frame. In particular, for \(\bar{v}\gg \Delta v\), the rate of energy exchange of the electrons is approximated as the product of the momentary interparticle Coulomb force and the centre-of-mass velocity in the laboratory frame (that is, \(P={F}_{\mathrm{C}}\bar{v}\)). The final energy difference then becomes ΔE = ∫dtP(t). Therefore, the nearly negligible initial Coulomb energy is magnified by the continuous centre-of-mass acceleration to a large final energy difference.

In Fig. 2e, the kinetic energy difference (magenta), interparticle Coulomb energy (purple) and momentary electron velocity (green) of the second particle are plotted on a double-logarithmic scale as a function of the distance of this particle from the emitter surface. It is evident that a few-hundred-millielectronvolt energy separation emerges on propagation to the extractor electrode in the electrostatic gun, while a further increase in the final energy difference of nearly 2 eV for these particles requires propagation and acceleration over several more millimetres. We note that by using a controlled femtosecond gate, this scenario represents a maximally controlled limit of the stochastic Coulomb interactions in the initial acceleration stages of electron microscopes67,68, first addressed by Boersch5 and Loeffler6.

The simulations also yield further insight into the characteristic timescales over which this electron–electron correlation persists. Figure 2f displays the computed final energy difference as a function of the initial temporal separation of two electrons (black solid line). We find that the energy difference drops to about 1 eV within 200 fs. As the laser pulse acts as a temporal gate for the photoemission, a prediction of the energy correlation function is obtained from these computed energy separations, weighted by the distribution of emission time differences under the photoemission laser pulse envelope (shaded area). The gap then arises from the fact that the laser pulse duration of ~150 fs does not lead to a substantial fraction of electron pairs with a larger separation in emission time. We experimentally probe this interpretation by conducting measurements using a pair of laser pulses of variable delays (Fig. 2g). The measured correlation gap closes for a temporal separation of the two laser pulses longer than 200 fs. Beyond such delays, an increasing number of electrons with small energy differences and near the central energy of the beam are found, and for those events, one electron is emitted in each pulse. A direct comparison of the experimental and simulated energy correlation functions for pulse overlap and for 0.4 ps two-pulse delays (Fig. 2h) yields convincing agreement. We also find excellent agreement with simulations for n = 3, 4 states (Extended Data Fig. 3).

Alongside their spectral distributions and correlations, the few-electron states observed here possess characteristic spatial properties. Specifically, in Fig. 3b, we show n-dependent beam caustics, which exhibit discrete differences in both the minimum spot size and focal position. Variations in the laser power yield changes in the caustics (higher power leads to some increase in spot size), but are far less pronounced than the differences between the event classes. Under the given conditions, the focusability is limited by spherical aberrations of the objective lens and the virtual source size, which result in typical spot profiles for positive and negative defocus (inset in Fig. 3b). The n > 1 caustics are evidently the result of a larger effective source, and the beam waist is shifted towards positive defocus.

Fig. 3: Characterization of the spatial electron beam properties.
figure 3

a, Schematic of the effect of Coulomb interactions on an electron beam coupled to an electron microscope. The virtual source increases in size and is shifted along the electron beam axis for pulse charges of two and three electrons. b, Caustics of the electron beam sorted by n, recorded by varying the last condenser lens of the microscope (light colours correspond to low power, dark colours to high power). The insets are images of the beam profile in underfocus (left), focus (middle) and overfocus (right) for n = 2 (blue outlines). c, Image of the n = 2-beam profile in underfocus, and the correlation angle φ between electron pairs with respect to the beam centre. The underfocus condition allows a precise measurement of the angular correlation. d, A strong anisotropic angular correlation is observed for n = 2 compared with an isotropic distribution for drawing random events from the n = 1 event class. The datasets employed are indicated by black circles around the data points in b.

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Both observations can be understood from mutual transverse deflection (sketch in Fig. 3a), which laterally spreads the few-electron trajectories17 such that the virtual source increases in size and moves forwards, as predicted in simulations36,38.

A more detailed view of the spatial properties of few-electron states is obtained by analysing correlations in the transverse momentum. To this end, we measure position correlations for a sufficiently large negative defocus (Fig. 3c). The spatial correlation is quantified via the angle φ between the two electrons and the beam centre. Figure 3d shows the angular correlation density of the two-electron state compared with random correlations drawn from a corresponding single-electron state at the same spot size (15 nm). In the electron pair state, we obtain a strong anisotropic correlation with a maximum around an angle of 180°, corresponding to electron events localized on opposite sides of the defocused beam, and thus exhibiting nearly opposite transverse momenta. Moreover, the angular correlation becomes most pronounced for events with the largest transverse momentum (Extended Data Fig. 4).

These observations clearly demonstrate that two-particle Coulomb interactions induce pronounced correlations in both the longitudinal and transverse momenta of the electrons. As the correlation primarily emerges in the initial acceleration stages of the electron gun, we explore the extent to which they can be controlled by the extraction fields. Qualitatively, a larger acceleration field is expected to enhance the longitudinal correlations and large kinetic energy differences in the beam direction, while a weaker acceleration allows the electrons to exchange more transverse momentum while limiting the growth of the final energy difference. Figure 4a sketches this trade-off between longitudinal and transverse correlations, which manifests experimentally in distinct properties of the few-electron states. Specifically, a decrease in the extraction voltage (that is, in the potential difference applied between the tip and the first anode) substantially reduces the observed energy correlation gap and the slope of the high-energy tail (Fig. 4b, solid lines; crosses in Fig. 4d denote the peak of the correlation function). Both features are reproduced in the two-particle simulations described earlier (Fig. 4b, dashed lines).

Fig. 4: Electric-field control of longitudinal versus transverse correlations.
figure 4

a, Via the extraction voltage, Uext, the initial acceleration serves as a control parameter to favour either energy or transverse momentum separation in the doublet states. Electron optical elements act as transverse momentum filters (background shading) and lead to lower transmission of electron pairs for weaker initial acceleration (low Uext). b, Electron pair correlation functions for varying extraction voltage. Particle tracking simulations (dashed lines) agree with the observed changes in correlation gap with Uext. The inset shows a schematic of the electron optics with emitter, extractor and aperture. c, Delay-dependent current–current correlation function g2(τ) for Uext = 1,900, 750, 400 V. The suppression at zero delay (g2(0)) is most pronounced for low Uext. d, g2(0) (left y axis) and energy correlation gap (right y axis) versus Uext.

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Interestingly, the enhanced transverse interaction at lower extraction fields has an immediate impact on the statistical distribution of the electron number states. Specifically, the measured beam caustics in Fig. 3b show that the convergence angle and thus the maximum transverse momentum of the individual particles is the same for all event classes, irrespective of n, primarily limited by the microscope’s condenser aperture. Therefore, the additional transverse momentum gained by Coulomb repulsion leads to a loss of total transmission of electron pairs. The corresponding change in the statistical distribution of events is expressed in terms of the second-order (current–current) correlation function g2(τ) as a function of the delay τ between recorded electrons, shown in Fig. 4c. We see that the emitted charge in sequential pulses is statistically independent (g2(τ) ≈ 1), while clear antibunching is observed for the electrons recorded from a single pulse (g2(0) < 1). In other words, at Uext = 400 V, the probability of detecting n = 2 electrons is reduced by a factor of 1 − g2(0) = 0.43 compared with a Poissonian process with the same average electron number per pulse. Importantly, we determine that the antibunching becomes much more pronounced for a smaller extraction voltage (Fig. 4d, circles), illustrating that the enhanced transverse correlations leads to a loss of pairs in the beam path by momentum-selective transmission. As in the case of the energy correlation, the controlled femtosecond temporal gate enabled by photoemission facilitates such strong antibunching22, which is orders of magnitude larger than has been observed for continuous20,38 and nanosecond-pulsed21 electron microscope beams.

Although the employed event-based measurements in conjunction with photoemission gating reveal important aspects of these few-particle correlations, it should be noted that the same phenomena will contribute to the properties of conventional (continuous) electron beams, with direct ramifications for the total beam brightness, coherence and non-correctable stochastic aberrations. However, in turn, the specific knowledge of these correlations allows control of the number statistics in the photoemitted beam, which may directly benefit microscopy applications. The antibunching observed here and in recent work22 implies that the total photocurrent exhibits sub-Poissonian noise characteristics—a property that is highly sought after in condensed matter scenarios (for example, as achieved by Coulomb blockade69). In the context of electron microscopy, this feature could be directly applied for shot-noise reduction in imaging and lithography, with immediate consequences for low-dose applications owing to the possibility of avoiding multi-electron specimen damage. In fact, our findings may be directly relevant for the mechanisms underlying the recently observed reduction in sample degradation with pulsed beams70,71. Further potential arises from the strong Coulomb correlations in energy and momentum identified for the few-electron states. For example, the fact that both electrons in the doublet state are well separated in energy and transverse momentum from each other allows for an energetic or spatial selection of the respective number state. This facilitates a powerful approach to controlling the statistics of single- and double-electron events.

In particular, the analysis of the measured spot profiles shows that a spatial aperture in a beam cross-over could be used to selectively favour the transmission T1 of the n = 1 number state by a factor of 3 and nearly 8 over the transmissions T2 and T3 of the n = 2 and n = 3 states, respectively (Fig. 5a,b). Similarly, a pre-specimen energy filter commonly used in state-of-the-art electron microscopes72 could be adjusted to enhance the transmission probability of n = 1 compared with n = 2 states (Fig. 5c). Specifically, for experimentally measured single-electron and double-electron spectra (Fig. 5e), the n = 1 transmission probability exceeds the n = 2 transmission probability by a factor of 8 at small slit widths, greatly amplifying the sub-Poissonian nature of the electron number distribution and facilitating a shot-noise-reduced electron current. Conversely, a central beam stop in energy could suppress a substantial fraction of single-electron states, leading to up to 20-fold enhancement of pair-state over n = 1 state transmissions (Fig. 5d,f). This approach will enable new forms of microscopy and spectroscopy with correlated electrons for a variety of two-point or two-time measurement schemes in correlated materials and free-electron quantum optics.

Fig. 5: Statistical control of single- and double-electron states using spatial and spectral filtering.
figure 5

a, A pinhole positioned at the beam waist of the n = 1 beam profile spatially filters higher-number states. b, State-selective beam transmissions Tn (bottom, calculated from data in Fig. 3) and transmission ratios T1/T2 and T1/T3, with increased relative selectivity of the n = 1 electron state (top). c,d, Left: in a spectrally dispersed plane, an energy-selective slit (energy width ΔEslit) reduces the transmission of n = 2 electron states (c), whereas an energy beam stop (energy width ΔEstop) suppresses the n = 1 electron states, enhancing the transmission of n = 2 electron states (d). Right: experimental spectra of n = 1 and n = 2 electron states (grey shading indicates the spectral density rejected by the energy slit/beam stop). e,f, Transmission Tn (bottom) and transmission ratios T1/T2 and T2/T1 (top) for the scenarios in c (e) and d (f). Considering individually optimized energy windows, 8-fold and 20-fold enhanced state selectivities are found for n = 1 and n = 2, respectively.

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In conclusion, the highly correlated electron number states introduced in this work are of interest both for fundamental considerations and their potential utility in manifold electron beam applications. For example, the pair state can be employed to implement a high-fidelity source of electron-heralded single electrons, enabling shot-noise-free imaging and lithography with a precisely counted number of electrons. Furthermore, the elementary scattering process that creates these well-defined few-body states might generally be assumed to induce entanglement between the electrons. Future studies may address the coherence of such multi-electron states and their possible use as free-electron qubits, with potential applications spanning from interaction-free or correlation-based quantum electron microscopy to quantum information processing.

Finally, we would like to note the study by S. Meier et al.73 on energy correlations of photoelectron pairs emitted from a free-standing tungsten tip, printed as a companion paper in this issue.

Methods

Femtosecond electron pulse generation in a transmission electron microscope

The experimental work was carried out in two commercially available transmission electron microscopes (JEOL JEM 2100F and JEM F200) that have been modified to allow the investigation of ultrafast dynamics in a stroboscopic laser-pump/electron-probe measurement scheme16. As our electron source, we employ W(100)/ZrO Schottky emitters (radius of curvature r = 490 nm, ~100 nm physical emission size) operated at Uext = 0.4–2.1 kV and a bias voltage of Ubias = −0.3 kV. After cooling the W(100)/ZrO emitter to just below the continuous Schottky emission threshold (filament current 1.6 A), the work function is close to the photon energy of the laser (Eph = 2.4 eV, corresponding to a 515 nm central wavelength). We generate ultrashort electron pulses via close-to-threshold linear photoemission by focusing laser pulses (160 fs pulse duration, 600 kHz/2 MHz repetition rates, 30 μm × 20 μm spot size) onto the apex of the nanotip. Apertures in the electro-optical beam path limit the transmitted beam to electrons that were generated close to the optical axis, resulting in average transmitted bunch charges of below one electron per pulse. Subsequent acceleration to 200 keV energies and coupling to the microscope column enables a pulse characterization in real and reciprocal space; spectral pulse properties are studied using an imaging energy filter (CEFID, CEOS).

Event-driven photoelectron detection

The correlated photoelectron states are imaged with a hybrid pixel electron detector based on the Timepix3 ASIC (EM CheeTah T3, Amsterdam Scientific Instruments) and mounted behind the imaging energy filter. The camera generates a stream of data packages containing the position of electron-activated detector pixels, their times of arrival (digitized with 1.56 ns time bins) and the energy (time-over-threshold, TOT) associated with incident electron events. At a beam voltage of 200 kV every individual electron activates a cluster of pixels of variable size (Npixels,avg ≈ 8 pixels), shape and energy (TOTavg ≈ 280 a.u.).

Single-electron-event localization of the TOT-corrected raw data stream is achieved using the Division of Nanoscopy, M4I, Maastricht University event clustering code74,75, which is based on a hierarchical density-based spatial clustering in Python3. The algorithm reconstructs the timing and position of individual electrons incident on the detector from the activated pixels. Individual electrons are thereby distinguished in terms of their times of arrival, attributing between three and nine neighbouring pixels activated within a time window of 100 ns and a summed TOT ranging from 200 a.u. to 400 a.u. for the same cluster (see ref. 74).

In a second step, the photoelectrons are clustered according to the femtosecond laser pulse that generated them. The temporal resolution of the detector (1.56 ns) is much faster than the temporal pulse separation given by the laser (500 ns and 1.6 μs for repetition rates of 2 MHz and 600 kHz), but much slower than the temporal splitting of the correlated electrons at the detector (~1 ps). The electrons arriving at the detector within Δtn = 50 ns are thus assigned to a number-class electron state n = 1, 2, 3, … determined by the number of electrons per laser pulse. The length of the electron correlation time window Δtn is chosen to capture all correlated electrons while being much shorter than the dead time between laser pulses (Supplementary Section A).

Effect of stochastic Coulomb interactions and the mean field on few-electron states

Even though only a fraction of electrons generated at the emitter surface is transmitted to the microscope column17, the spatiotemporal confinement of the emission results in a non-negligible influence of the entire electron cloud on the properties of the transmitted beam. Consequently, mean-field (space charge) and stochastic interactions between all electrons both need to be considered and distinguished from the correlations observed in the electron pair state. These different contributions can be assessed by laser-power-dependent measurements. The corresponding n = 1 and n = 2 spectral distributions, as well as the n = 2 average pair energy (EA + EB)/2 (Extended Data Fig. 2a,b,d), display the expected broadening with increasing laser power (compare Extended Data Fig. 2e; n = 1 broadening (orange circles) and average pair energy broadening (grey circles)) that scales with the average photocurrent. This is in close correspondence to previous non-event-selective measurements17,35,65 and is typically attributed to stochastic Coulomb interactions and mean-field effects.

In contrast, the two-electron correlation functions displayed in Extended Data Fig. 2c are remarkably independent of laser power, showing a pronounced gap that is about 1 eV wide, a peak at around 1.8 eV and an extended tail towards large energy separations exceeding 4 eV. Increasing the photocurrent only imposes moderate variations in the depth of the gap and the shape of the high-energy tail. In particular, the position of the main correlation peak (Extended Data Fig. 2e, blue circles) approaches a fixed value of 1.7 eV towards small average currents, demonstrating that the observed correlation is only weakly altered by multiple Coulomb interactions with the space-charge cloud. The peak position is instead dominated by the two-electron correlation.

State-averaged energy-subtracted spectra

Shot-to-shot variations between electron pulses deteriorate the state-averaged (Extended Data Fig. 1a) and number-state resolved (Extended Data Fig. 1b–e) spectra. They are primarily caused by high-voltage and space-charge fluctuations that change \(\overline{E}\). As a result, the characteristic multi-peak spectra of the few-electron states are blurred, particularly for electron states with n ≥ 2. Correcting every individual pulse for \(\overline{E}\) thus substantially enhances the visibility of the multi-peak spectra (compare Extended Data Fig. 1f–h). The root-mean-square widths of the state-average energies shown in Extended Data Fig. 1i–k are reduced for higher number states (n = 2, 3, 4: 0.73, 0.6, 0.52 eV).

Two-laser-pulse electron generation

For the two-laser-pulse generation described in Fig. 2g, a Michelson interferometer splits the incoming laser pulse into two separate pulses. One of the interference arms has a variable optical path length, implemented by a retroreflector mounted on a delay stage. The delay time of the two optical pulses (up to 2 ps) is much shorter than the laser pulse repetition time (1.6 μs, corresponding to a repetition rate of 600 kHz). Hence, two photoelectrons generated by two separate laser pulses and two photoelectrons generated by the same pulse are both detected as two-electron events.

As the optical power on the tip oscillates for small delay times due to constructive and destructive interference of the laser pulses, the number of generated electrons strongly varies in this delay regime. Therefore, we select delays with approximately the same one-electron-state rate (±σ/2) over the integration time of 5 s (see Supplementary Section B for a detailed description of the data selection).

Numerical simulations of multi-particle trajectories

Energy correlation histograms for the electron number states n = 2–4 are shown in Fig. 2b–d. These correlation spectra are reproduced with the numerical multi-particle trajectory simulations discussed in Fig. 2e–h and in Supplementary Section C. For the simulation of the n = 3, 4 correlation spectra, the model is extended to three and four particles. We compute the electron trajectories of all n states for a set of parameters within the experimental range: an extraction voltage of 2,100 V, a temporal emission profile of 180 fs, a physical source size of 100 nm and considering the mean-field broadening of 1 eV observed for the n = 1 state. The simulated multi-particle energy-pair histograms are shown in Extended Data Fig. 3d–f and are in excellent agreement with the experimental data (Extended Data Fig. 3a–c) in terms of the observed correlation gaps and peak positions.