Electrons herald non-classical light

Abstract

Free electrons are a universal source of electromagnetic fields, and fundamentally their quantized energy exchange may facilitate generating tunable quantum light. Because the quantum features of the emitted radiation are encoded in the joint electronic and photonic state, they can only be revealed by a measurement accessing both subsystems. Here we demonstrate the coherent parametric generation of such non-classical states of light by free electrons. Investigating electron–photon correlations, we show that the quantized electron energy loss heralds the number of photons generated in a dielectric waveguide. In Hanbury Brown–Twiss measurements, we observe an electron-heralded single-photon state using antibunching intensity correlation, whereas two-quantum energy losses of individual electrons yield pronounced two-photon coincidences. Our results will enable the tailored preparation of higher-number Fock and other optical quantum states on the basis of controlled interactions with free-electron beams.

Main

Quantum states of light facilitate manifold applications in communication¹, computation² and sensing³. As a hallmark of quantum optics, photon number states have no classical counterpart and are essential in quantum metrology⁴ and information processing⁵. Multiple approaches and platforms have been used for their generation, including circuit quantum electrodynamics systems^6,7,8 and superconducting structures⁹ in the microwave regime, as well as spontaneous parametric downconversion^10,11,12 in the visible to near-infrared.

In the search for new sources of non-classical light, free electrons present an attractive pathway. In pioneering works, electron-excited single-quantum emitters, such as colour centres and point defects, have already shown antibunching photon statistics in incoherent emission^13,14,15. Parametric photon generation by coherent cathodoluminescence^16,17,18, on the other hand, promises sophisticated photonic states without involving Fermionic excitations, preserves the quantum phase between electrons and photons¹⁹ and allows for enhancements by tunable phase matching^{20,21,22,23,24}. In this context, photonic integrated circuits^25,26 offer low-loss photon transport, as well as versatile mode tailoring, facilitating efficient scattering between electrons and optical modes²⁷. Combining free electrons and photonic integrated circuits thus provides a unique platform for the excitation, control and observation of classical and quantum optical states at widely variable wavelengths.

A key element of realizing electron-driven quantum light will be to harness the correlations and entanglement of photonic and electronic states. In particular, electron–photon coincidence detection^{23,28,29,30,31} has yielded detailed information on the joint electron–photon state in time, energy and momentum. Post-selection of time-correlated electrons and photons was recently shown to improve X-ray elemental mapping²⁹, as well as photonic-mode imaging and spectroscopy^23,30. Furthermore, a temporal fingerprint of the interaction may yield material- and excitation-specific decay times^32,33 and improve the signal-to-noise ratio in electron microscopy^23,34. Coherent cathodoluminescence combined with post-selection is further expected to enable the preparation of electron-heralded Fock^35,36, Gottesman–Kitaev–Preskill (GKP)^37,38 and other complex quantum states of light^37,39. However, the preparation of non-classical optical states via parametric electron–photon interactions has not yet been shown.

Here we demonstrate the generation of electron-heralded non-classical light using spontaneous parametric scattering in an integrated photonics waveguide. We merge energy-resolved single-electron detection with two-photon coincidence measurements in an optical Hanbury Brown–Twiss (HBT) set-up to probe strong correlations between the generated photon number and the quantized electron energy loss. Electron heralding to a single-photon energy change yields prominent photon antibunching. The high-contrast detection of threefold coincidences evidences two-photon generation, establishing a platform for heralded photon number states in integrated photonics-based quantum technology.

In our study, electron-induced photon generation takes place at a Si₃N₄ waveguide (2.2 μm × 780 nm cross-section, geometry as seen in Fig. 1a,d, embedded in SiO₂) on a photonic chip, which is inserted into the sample plane of a transmission electron microscope (TEM) (cf. Fig. 1a). A weakly focused continuous electron beam (100-keV energy, 30-nm beam diameter, 1.1-mrad half angle) passes the waveguide parallel to its surface at an impact parameter (distance from the surface) of about 250 nm (Fig. 1b). Mediated by the structure’s dielectric response, the electrons at initial energy E₀ interact with the optical vacuum of the waveguide modes, generating photons in the modes in a parametric process^18,23,40. As in optical wave-mixing and spontaneous parametric downconversion (SPDC), ‘parametric’ refers to a direct and quantum-coherent energy exchange between the electrons and the guided optical mode, leaving no auxiliary excitations in the dielectric material. This entails instantaneous emission and electron–photon energy and momentum conservation, as well as coherent amplitude build-up in the phase-matched interaction. Recently, the observation of Ramsey-type interference at a photonic ring resonator could also verify the coherence of the photon generation and propagation²³.

Fig. 1: Correlation measurement of waveguide-coupled photons and free electrons.

a, A beam of electrons passes a dielectric waveguide, generating photons that are guided and detected in a fibre-based HBT set-up. The electron and photon arrival times are temporally correlated. b, Enlarged sketch of the interaction region: the electron passes the waveguide surface at an impact parameter of around 250 nm to interact with the evanescent vacuum field of the waveguide modes. c, Simulation of the photon spectral distribution generated in a 40-μm straight waveguide by 100-keV electrons (impact parameter 250 nm). Grey vertical lines: waveguide boundaries. a.u., arbitrary units. d, False-colour image of an example photonic chip (scanning electron micrograph). The chip surface is covered with ITO (indium tin oxide; brighter surface, ITO on metal; darker surface, ITO on Si₃N₄ or on SiO₂; side walls, SiO₂). e, Schematic of the correlated quantized electron energy loss and photon number. The electron scatters at the optical mode, generating a random number of photons. After detection, the signal correlation links the photon number k to the electron scattering order m. The generation process is expected to follow a Poissonian distribution.

The interaction can, therefore, be described as inelastic electron–photon scattering, without material excitation, that correlates and entangles the final electron energy ${E}_{0}-{\sum }_{i = 1}^{k}\hslash {\omega }_{i}$ with k generated photons at frequencies ω_i. In this situation, the frequencies lie within a spectral continuum of possible excitations (scattering operator given in Supplementary Section 1)⁴¹ defined by a phase-matching condition in which the electron group velocity is equal to the photon phase velocity (Supplementary Section 4)^35,40. It is worth noting that this phase-matching, in contrast to SPDC, is simultaneously satisfied for all photon orders. This follows from the fact that scattering with one or more photons does not substantially alter the velocity of the electron or its path during the passage of the structure (non-recoil condition^18,42). Hence, one electron passing through the optical mode can produce multiple photons sharing identical properties.

In an effective single-mode description, the electron–photon state resulting from direct, parametric electron–photon interaction can be simplified to

$$\left\vert {\psi }_{\mathrm{e}},{\psi }_{\mathrm{ph}}\right\rangle =\mathop{\sum}_{k}{c}_{k}\left\vert {E}_{0}-k\hslash {\omega }_{0},k\right\rangle .$$

(1)

Here, the coefficients

$${c}_{k}\approx {\mathrm{e}}^{-| {g}_{0}{| }^{2}/2}{g}_{0}^{k}/\sqrt{k!},$$

(2)

correspond to a Poissonian process of quantized energy exchanges producing k photons near a central optical frequency ω₀, with a dimensionless coupling constant g₀ = √(∫dω |g_ω|²) obtained by integration over the spectral coupling strength density g_ω. For the designed structure, the photons are predominantly detected from the quasi-TM₀₀ (transverse magnetic) mode at wavelengths between 1,300 and 1,400 nm (see simulation results in Fig. 1c), while other excited optical modes are suppressed (Methods). The parametric generation process renders the photons phase-coherent to their respective electrons, resulting in so-called coherent cathodoluminescence^16,17, which yields an optical state with the above-mentioned Poisson statistics. As the arrival time of subsequent electrons is random, however, the optical state produced is super-Poissonian in nature. The generation of a coherent state from multiple electrons would require electron bunching or electron energy combs^19,37,43,44.

The induced optical state is analysed by coincidence detection of the generating electrons and the created photons (Fig. 1b). While the photons are guided through fibres to optical single-particle detectors, the electrons are dispersed in a magnetic spectrometer and recorded with an event-based detector (Timepix3 logics), which delivers the kinetic energy and arrival time of each electron. The readout electronics of the electron camera further provide time stamps for the arrival time on both photon detectors for electron–photon–photon temporal correlations. We expect to detect different k in each parametric electron scattering event, with the k photons being temporally linked to electron energy shifts of mℏω₀ (Fig. 1e), with m the electron scattering order. For an ideal electron–photon state (equation (1)), the photon number and quanta of electron energy change are identical (k = m), with probabilities following c_k². In this case, combining temporal particle correlation and electron-energy analysis enables heralding schemes, in which the detection of an electron in a specific energy state |E₀ − kℏω〉 unambiguously heralds the photon number state |k〉. In practice, finite optical transmission and inelastic scattering that does not inject light into the waveguide will give the quantized electron-energy loss m an upper bound k ≤ m; to reach this will be a technological target.

We gain access to single- and two-photon generation events and characterize the optical state via an electron-heralded optical HBT detection set-up⁴⁵ shown in Figs. 1a and 2a. Two single-photon detectors A and B placed behind a 50/50 beam splitter are synchronously time-stamped by the electron detector, and every electron is assigned to the closest photon on either detector, calculating the time delay between them. The analysis of the resulting electron–photon–photon correlation counts N_e,A,B is conducted as a function of the electron-energy loss E_el and the relative time delays τ_A = t_A − t_el and τ_B = t_B − t_el between the electron and photon detection (arrival times t_el, t_A and t_B). An event histogram over these degrees of freedom results in a data cube, which we investigate along cuts or in projection on different axes (Fig. 2b).

Fig. 2: Three-particle correlation for one electron and two photons.

a, Schematic of the detection set-up with the particle arrival times t_el, t_A, t_B on the fibre-coupled photon detectors D_A, D_B and the electron detector D_el and electron energy E_el as measured parameters. The arrival times are correlated with each other. b, Parameter space for threefold events. The data are analysed through linecuts and integrals along different axes. c, Histogram of electron–photon–photon coincidence events N_e,A,B as a function of τ_A and E_el, summed over τ_B. Electron–photon scattering causes a coincidence peak at an electron energy shifted downward by around 0.9 eV or 1 ℏω. Top: the temporal coincidence uncertainty of around 2.9 ns FWHM represents the timing precision of electron detection. d, Histogram of particle triples as a function of t_B − t_A and E_el filtered to true coincidences between a photon A and an electron. The coincidence peak is now downshifted from the unscattered electrons by 2 ℏω. The coincidences are, therefore, dominated by electrons that generated two photons. Top: the linecut shows a higher resolution of the correlation peak due to the smaller photon detector jitter.

The electron–photon correlation as a function of τ_A and E_el is given in Fig. 2c. The correlation exhibits a strong coincidence peak separated from the uncorrelated, mostly unscattered electrons by approximately 0.9 eV (corresponding to a free-space photon wavelength of λ = 1,350–1,420 nm). Coincidences are found within a temporal interval of 2.9 ns (FWHM, full-width at half-maximum), mainly defined by the electron detection jitter δt_el (ref. ²³). Above a constant background in τ_A, the temporally confined coincidence peak consists of events in which at least one electron-induced photon was generated, and both a photon and its generating electron were detected. The detected electron–photon coincidences primarily originate from the excitation of the detectable TM₀₀ mode. Higher-order modes are suppressed in the chip–fibre coupling, and the weakly excited TE₀₀ (transverse electric) mode is further diminished due to the detector cutoff wavelength (Methods).

Electron-heralded two-photon generation can be observed via a photon A–photon B correlation trace, filtered to true coincidences between photon detector A and an electron (Methods). The arising two-dimensional coincidence histogram in Fig. 2d exhibits a strong two-photon correlation peak (1.8 eV = 2 ℏω energy loss) above a weak background (less than 5% of the peak), in which the three detected particles did not all stem from one scattering event (generating electron undetected at m = 0 and zero delay, and one improper random photon detection at m = 1). The photon–photon coincidence peak has a width of ~400 ps (FWHM, δt_ph < 300 ps photon detector jitter).

The inelastic electron scattering at the sample is visualized in the electron energy spectra displayed in Fig. 3. The uncorrelated spectrum (Fig. 3a) contains all channels of inelastic electron scattering⁴⁶, including material-specific radiative and non-radiative effects and quasiparticle excitation⁴⁷ as well as the desired coherent photon generation into optical modes. The electron spectrum (Fig. 3a, blue) is modelled (red) by the convolution of the comb-like coherent cathodoluminescence peaks, following a population distribution described by P_m = ∣c_m∣² (see equation (2) with k = m), with dielectric losses at the waveguide material, exhibiting an exponential decay with energy¹⁸ (for more details, see Supplementary Section 6). Removing the effect of dielectric losses, the spectral component governing coherent photon generation is given by the yellow curve, with a peak-to-peak distance between the sidebands close to 0.9 eV. The overall coupling constant of the electrons to the waveguide optical modes ${g}_{0}^{{{\rm{EELS}}}}$ can be calculated from the population distribution of the individual sidebands P_m (Fig. 3a, violet to light-blue peaks). In the experiment, some averaging of the coupling constant is observed, caused by a broad distribution of impact parameters due to a non-zero convergence angle and finite beam-focus diameter (Fig. 3b,c), as well as small time-dependent beam shifts. Using the population ratio between the first three sidebands in the electron spectra, we obtain an average coupling constant $\langle {g}_{0}^{{{\rm{EELS}}}}\rangle =0.32$ with an s.d. of ${\Delta }_{{g}_{0}^{{{\rm{EELS}}}}}=0.24$ (see Supplementary Section 6.2 for calculations).

Fig. 3: Multiple scattering and heralded electron spectra.

a, Time-averaged (not photon-correlated) electron spectrum (blue), modelled (red) by cascaded m-photon multiple-loss peaks (yellow) and an exponentially decaying continuum. b, Sketch of the electron-beam caustic traversing the waveguide with a convergence half-angle of α = 1.1 mrad, causing some spatial averaging of the coupling constant. c, Experimentally observed decay of the mean coupling strength towards larger impact parameters (distance from the waveguide). The s.d. is given as a grey background. The peak-height distributions at different ${g}_{0}^{{{\rm{EELS}}}}$ are included as insets. d, Energy distributions of uncorrelated (blue) and correlated electrons (red, yellow) for k detected photons (normalized to the area under the curves). The correlated spectra are obtained by summing over the background-subtracted coincidence peaks shown in Fig. 2c,d. e, Measured optical spectrum with a central peak width of about 80 nm, or 65 meV, normalized to the maximum value.

Whereas the unheralded electron spectrum includes all inelastic scattering pathways, access to the properties of the detected optical mode is enabled through electron–photon coincidences. The spectra of electrons in coincidence with one (red) or two (yellow) detected photons (Fig. 3d) closely resemble copies of the uncorrelated spectrum (blue) when normalized to the area under the curve. The central wavelength of the TM₀₀ mode, which dominates the transmitted photons, defines the relative shift of the spectra of around 0.9 eV. A moderate spectral broadening of 65 meV per electron scattering order can be attributed to the phase-matching bandwidth of 80 nm with the optical mode (Fig. 3e). This corresponds to an effective interaction length of around 40 μm, in which the electron beam and the optical mode’s field can be regarded as parallel (see simulation results in Supplementary Section 4). Generally, the probability of photon generation (~g₀²) scales linearly, and the optical spectrum’s width scales inversely with the waveguide length. Overall, these results are fully consistent with the underlying parametric scattering process. In particular, the electron energy loss agrees with the optical emission wavelength, the phase-matching simulation predicts the full optical spectrum and the photon generation is instantaneous within the measurement precision (Fig. 2).

While the previously calculated ${g}_{0}^{{{\rm{EELS}}}}$ includes all excitable modes, we are primarily interested in the strong coupling to a single optical mode. We gain access to its coupling constant via the intrinsic mode filtering of the optical set-up (Methods) and subsequent electron–photon coincidence detection. The coupling strength g₀ of electrons to the detected TM₀₀ mode is determined taking into account optical losses (equation (2), see Supplementary Sections 5 and 7 for further details). The detection probabilities of one or two photons P₁ and P₂ are deduced from the area of the m = 1 and m = 2 sidebands in the electron–photon and electron–photon–photon coincidence spectra, respectively. They define the number of single- (two-) photon excitation events in which one (two) photons were detected. The calculated g₀ > 0.2 is somewhat lower than the value extracted from the electron spectrum, illustrating the necessity of coincident photon detection for disentangling multiple scattering pathways and, in particular, for isolating coherent photon generation.

Beyond the spectral characterization of heralded and unheralded electronic states, we quantitatively study the electron–photon–photon coincidences. By filtering electron–photon coincidence events to electron energies of E₀ − mℏω, the generated optical state is expected to contain m photons, rendering the heralded state non-classical. The photon statistics of the measured optical state are obtained via (heralded) intensity correlation g⁽²⁾, a measurement achieved through an optical HBT scheme^45,48. Figure 4 analyses the relative-time histogram N_e,A,B(τ_A, τ_B, E_el), which counts threefold correlations at delays τ_A and τ_B in a defined electron energy window E_el. Heralding a photonic state by a specific energy change is achieved by integrating over a 540-meV wide energy window across the single- (m = 1, Fig. 4a) and two-photon (m = 2, Fig. 4b) energy-loss peaks. True coincidences of electrons with one photon detector N_e,A∨B (that is, at detector D_A OR D_B) lead to vertical/horizontal bands at τ_i ≈ 0, while photon–photon coincidences N_A∧B (excitation of detectors D_A AND D_B) with or without a corresponding electron emerge on the diagonal line along τ = τ_A − τ_B ≈ 0. The temporal widths of the coincidence features follow the respective detector jitters (Fig. 2c,d). Threefold coincidences are found at the intersection of the two features. While m = 1 shows electron–single-photon coincidences and some photon–photon coincidences not linked to an electron, the plot for m = 2 is dominated by two photons in coincidence with a detected electron. This confinement of three-particle coincidences to the 2-ℏω loss region (Figs. 2d and 3d) demonstrates the generation of two-photon states via parametric electron scattering with a two-photon energy loss. Photon–photon coincidences on the diagonal line off the centre arise from two-photon generation events assigned to a different electron than the generating one, which went undetected.

Fig. 4: Statistical analysis of generated photons.

a,b, N_e,A,B as a function of photon-electron time delays, selected by electron energy: m = 1 (a) and m = 2 (b). c, Violation of the Cauchy–Schwartz inequality (γ > 1) for the electron–photon interaction at different τ. d, Unheralded photon–photon intensity correlation for varying electron current. e, Electron-heralded photon–photon intensity correlation filtered to the energy region m = 1 as a function of both time delays. Inset: time-averaged g⁽²⁾ for m = 1. f, Photon–photon intensity correlation as a function of the number of photon-heralding electrons q between the two heralded photons without energy selection (top) and for m = 1 (bottom).

To test the non-classicality of the radiation process, that is, whether it is incompatible with classical field theory, we invoke the Cauchy–Schwarz inequality⁴⁹ $\gamma =\frac{{g}_{E,{{\rm{A}}}\vee {{\rm{B}}}}^{(2)}(\tau )}{{g}_{E}^{(2)}{g}^{(2)}(0)}\le 1$, assuming a classical radiation process. The inequality (for derivations, see Supplementary Section 2) bounds the temporal cross-correlation between the detected electron energy and photon number ${g}_{E,{{\rm{A}}}\vee {{\rm{B}}}}^{(2)}(\tau )$ to the variation of the electron energy loss ${g}_{E}^{(2)}=\frac{\langle \Delta {E}_{\mathrm{el}}^{2}\rangle }{{\langle \Delta {E}_{\mathrm{el}}\rangle }^{2}}$ and the temporal intensity correlation of photons in an HBT set-up g⁽²⁾(τ), normalized to τ ≫ 0. For electron–photon coincidence, we find a pronounced violation of the Cauchy–Schwarz inequality, meaning γ > 1 (Fig. 4c), which proves the non-classical nature of the process^50,51,52.

The photon statistics of this radiation is further examined using the temporal intensity correlation g⁽²⁾(τ) of detectors placed in a HBT set-up (Methods). While classical light shows g⁽²⁾(0) ≥ g⁽²⁾(τ ≫ 0), this is not the case for non-classical light such as single photons or higher photon number states. The unheralded photon–photon intensity correlation (taken with higher time resolution, higher-efficiency detectors and a time tagger, Methods) shows a bunching peak at time delay τ = 0 ps (ref. ⁵³), inversely proportional to the electron current (Fig. 4d) and generally larger than for thermal light, which features g⁽²⁾(0) ≤ 2. Following a current-dependent bunching height g⁽²⁾(0) ≈ 1 + 1/(Iτ_bin), where I is the electron flux rate and τ_bin is the temporal data bin width, this behaviour is expected when single electrons can create multiple photons, either in a direct process, as demonstrated here (see Supplementary Section 3 for further calculations), or as a result of cascaded excitation of multiple two-level systems leading to incoherent emission^15,54,55,56. The emergence of coherent photon statistics (g⁽²⁾(0) = g⁽²⁾(τ ≫ 0)) that result from the parametric photon generation process can be achieved through electron bunching or electron energy combs^19,37,43,44.

The statistics of the non-classical optical states heralded by energy-filtered electrons (540-meV energy window around a defined loss peak m) are analysed using the normalized heralded intensity correlation (Methods and Supplementary Section 8)

$${g}_{\mathrm{H}}^{(2)}({\tau }_{{{\rm{A}}}},{\tau }_{{{\rm{B}}}},{E}_{{{\rm{el}}}})=\frac{{N}_{{{\rm{e}}},{{\rm{A}}},{{\rm{B}}}}({\tau }_{{{\rm{A}}}},{\tau }_{{{\rm{B}}}},{E}_{{{\rm{el}}}})}{{N}_{{{\rm{e}}},{{\rm{A}}}}({\tau }_{{{\rm{A}}}},{E}_{{{\rm{el}}}}){N}_{{{\rm{e}}},{{\rm{B}}}}({\tau }_{{{\rm{B}}}},{E}_{{{\rm{el}}}})}{N}_{{{\rm{e}}}}({E}_{{{\rm{el}}}}).$$

(3)

This correlation normalizes the occurrences of N_e,A,B (Fig. 4a,b) to their single-photon equivalents N_e,A/B and the number of heralding electrons N_e. Figure 4e displays ${g}_{\mathrm{H}}^{(2)}({\tau }_{{{\rm{A}}}},{\tau }_{{{\rm{B}}}},{E}_{{{\rm{el}}}})$ for single-photon electron-energy losses (m = 1), as well as the so-called time-averaged heralded intensity correlation${\overline{g}}_{\mathrm{H}}^{\;(2)}(\tau ,{E}_{{{\rm{el}}}})$ (refs. ^57,58), which is reduced to the relative photon–photon delay (inset; see Methods and Supplementary Section 8 for equation and calculation). Their features are governed by N_e,A,B(τ_A, τ_B, E_el), discussed before (Fig. 4a). At τ_A∧B ≈ 0, g_H⁽²⁾ achieves near-zero values (inset) and only weak contributions from electron-heralded photon–photon coincidences are visible on the diagonal. Hence, for m = 1, two photons are rarely detected at similar times. Generally, in photon-only coincidence experiments, accidental and true heralded photon–photon coincidences cannot be distinguished in the heralded intensity correlation if the coherence time is lower than the respective detector jitters^58,59 (see also Supplementary Section 8). However, in our case, the electron-heralded photon–photon coincidences remain distinct due to the much better temporal resolution of photon detection compared with the heralding electron detector.

A discrete version of the intensity correlation function is given by

$${g}_{\mathrm{H}}^{(2)}[q,{E}_{{{\rm{el}}}}]=\frac{{N}_{{{\rm{e}}},{{\rm{A}}},{{\rm{B}}}}[q,{E}_{{{\rm{el}}}}]{N}_{{{\rm{e}}}}({E}_{{{\rm{el}}}})}{{N}_{{{\rm{e}}},{{\rm{A}}}}({E}_{{{\rm{el}}}}){N}_{{{\rm{e}}},{{\rm{B}}}}({E}_{{{\rm{el}}}})}.$$

(4)

The number of coincidences N_e,A/B between electrons and photons A or B are counted by considering detection within a relative time window of ±2.6 ns (95% coincidence interval in Fig. 2c). Compared with the time-averaged intensity correlation, the photon–photon time delay τ is replaced by the number of heralding electrons q, by which photon arrivals on detector A are shifted relative to B for cross-correlation. This is similar to the number of pulses between two detected and correlated photons in pulsed SPDC. Thus, N_e,A,B[q, E_el] counts the coincidences between heralded photons A and heralded photons B that are separated in time by q electrons. The case q = 0 describes coincidences between two photons heralded by the same electron. By definition, g_H⁽²⁾ approaches unity for large ∣q∣ (sketch and further explanation given in Supplementary Section 8). The resulting intensity correlations are shown in Fig. 4f. Without energy filtering (top), a small bunching peak is observed with g_H⁽²⁾[0] = 1.10, similar to the unheralded intensity correlation function (Fig. 4d), but with larger bins corresponding to the electron detector resolution. Filtering to a single-quantum energy exchange (m = 1, same energy window as for Fig. 4e) leads to a strong antibunching behaviour with g_H⁽²⁾(0) suppressed to 0.056. The pronounced antibunching dip observed for both the time-dependent and discrete intensity correlation functions confirms the high purity of the heralded single-photon state. Two-photon states should, under ideal conditions, also show prominent antibunching. At present, however, technical aspects, including optical losses, limited electron spectral resolution and the statistics of the electron source, limit the ultimate purity of this heralded state (Supplementary Section 9). Such technical limitations can be improved by time-gated electron sources and enhanced sample designs.

The demonstrated scheme can be quantitatively analysed in its ability to serve as an electron-heralded photon-number source by calculating the electron-energy-specific intrinsic heralding efficiency ${\eta }_i^{I}$ and the coincidence-to-accidental ratio (CAR) (Methods). For heralding single photons (m = 1), we achieve ${\eta }_{{{\rm{A}}}\vee {{\rm{B}}}}^{I}=10 \%$ and CAR > 30, whereas photon pairs (m = 2) yield ${\eta }_{{{\rm{A}}}\wedge {{\rm{B}}}}^{I}=0.3 \%$ and CAR > 150, confirming that detected two-photon states are generated predominantly by double loss of single electrons, while single photons are generated by single-loss electrons. Experimentally, these values are limited by the spectral mixing of the heralding electron and the slight multimode character of the generated photons, and are expected to improve for an enhanced energy resolution⁶⁰, as well as improved coupling ideality to the detected mode⁴¹. Fock state generation further requires photon indistinguishability. While two photons generated by two different electrons will remain distinguishable by their electron degrees of freedom, multiple photons generated by a single electron are anticipated to be indistinguishable (for details, see Supplementary Section 10). The required identical scattering conditions are enabled by the use of high-energy electrons and a single excited optical mode. Future experimental proof of this property, enabled by tailored structure design, will further enhance the usability of the scattering scheme for versatile photon Fock state generation.

Quantifying our scheme in terms of electron-heralded photon generation requires a separation between different electron scattering pathways. We observe inelastic electron scattering at the material, as well as parametric photon generation. The unheralded electron spectrum samples a range of coupling constants reaching ${g}_{0}^{{{\rm{EELS}}}} > 0.5$ (Fig. 3c), slightly smaller than values reported for extended planar structures²⁴. Notably, in the present experiments, a large fraction of generated excitations populate a single broadband spatial mode. With a value of g₀ > 0.2, we report the highest coupling of electrons to a detected photonic mode so far. Reaching the strong-coupling regime by establishing unity coupling constant to a single optical mode^{35,41,61,62,63} will give direct access to a wide class of phenomena, such as efficient electron–photon entanglement^64,65,66,67, photon-induced electron–electron correlations^19,68 and the generation of more complex non-Gaussian optical states³⁹. Future work will also investigate the influence of decoherence in electron scattering at photonic modes, with first experiments already demonstrating electron–photon entanglement in different basis sets^69,70.

In conclusion, we introduce electron-heralded quantum states of light, identifying hallmark features such as antibunching photon statistics and multiphoton–electron correlations. The scheme will facilitate future electron-heralded high-order Fock state generation, on the basis of further technical optimizations and strong single-mode coupling. Combined with photon or electron state operations, the generation of a wide variety of complex quantum states comes into reach. Broadening the scope of free-electron quantum optics, our results experimentally establish electron beams as a tailored quantum optical resource.

Methods

Fibre-coupled integrated waveguides

The photonic chip with a Si₃N₄ waveguide embedded in SiO₂ is fabricated via the Damascene process²⁶ on a Si substrate. The waveguide (width × height 2.2 μm × 780 nm) has a separation between the bus waveguides, which connect the interaction region with the chip-to-fibre couplers, of around 125 μm. The longest path at which electrons can intersect with the waveguide is around 80 μm long. However, the curvature of the waveguide, as well as slight angular deviations between the waveguide and the electron beam, reduce the effective interaction length to around 40 μm. The waveguide has no top cladding, allowing for the optical modes to reach into the vacuum and interact with the electron beam. An ITO layer of <20 nm thickness (index of refraction n = 1.99 at 1,550 nm measured for ITO films deposited under similar conditions) on top of the waveguide reduces charging in the presence of an electron beam. The chip is mounted on a custom-made TEM holder²⁷ such that the waveguide sits in parallel to the electron beam. The optical outputs of the symmetrically built chip are connected to two short pieces of ultrahigh-numerical-aperture (UHNA-7) fibre, which are spliced to single-mode fibres (SMF-28). The single-mode fibres are sent through the hollowed-out TEM holder and a custom vacuum fibre feedthrough. As the photon generation is directional, the optical detection set-up is spliced to the SMF-28 fibre, which is connected to the lower output port.

The main excited optical mode of the waveguide is the quasi-TM₀₀ mode due to better phase-matching. Simulation results predict that, if the electron beam is placed centrally above the waveguide at an impact parameter of 250 nm, over 92% of the photons will be generated in this optical mode. Further modes are the TE₀₀ and higher-order modes. While the TE₀₀ mode only shows a small scattering probability in simulations, due to a small fraction of the electric field parallel to the electron beam, and the phase-matching energy lies in the detection cut-off region, the higher-order waveguide modes are suppressed during their transmission through single-mode optical fibres. TM₀₀ and TE₀₀ show transmission efficiencies around 20% and 25% respectively at the fibre connections. This effectively leads to a spatial single-mode optical detection.

TEM set-up

The experiments are conducted at the Göttingen UTEM (JEOL JEM 2100F) with a Schottky field-emission electron source operated in a continuous (thermal-field) mode. The resulting electron beam shows an energy spread of around 0.6 eV at an electron energy of 100 keV. The current is reduced to around 1.4 pA by adjusting the emitter temperature and by using a condenser aperture of 40 μm diameter to not oversaturate the electron detector. The experiments are conducted in low-magnification STEM (scanning transmission electron microscopy) mode, enabling the positioning and scanning of the focused electron beam (30-nm beam diameter, 1.1-mrad half convergence angle) in the sample plane. The scanning resolution is of the order of 30 nm. All measurements described in the main text are conducted with a stationary electron beam, which is checked and manually readjusted for an optimal combination of photon generation rate and electron transmission onto the detector every 10–30 min. Transmission losses in the electron channel arise from charging effects along the waveguide surface, distorting and shifting the electron beam, which reduces incoupling into the 5-mm spectrometer entrance aperture. Post-measurement calculations show an electron transmission efficiency of around 60–70%. For the scanned data, the electron beam is scanned multiple times across the same region (width × height 20 nm × 500 nm, 300-ms integration per pixel, ~40-nm pixel size). The scanned data are used to measure the position-dependent change of the coupling constant.

Optical set-up

The generated photons are sent through the waveguide and optical fibres to the outside of the TEM holder. The fibre end is spliced to a 50/50 beam splitter (Thorlabs TW1550R5A1), which distributes the photons to two cooled avalanche photodiodes (ID Quantique ID230). As the generated light is measured to be around 1,300–1,400 nm, we assume slightly higher losses and a deviation from the 50/50 coupling ratio at the beam splitter. The count rates indicate a splitting ratio of 52/48. The avalanche photodiodes are set to >25% detection efficiency and 20 μs dead time, which leads to 250 and 300 photons s⁻¹ intrinsic dark counts, respectively, for diodes A and B. An evaluation of the transmission and detection efficiency shows an approximate efficiency of 2% per diode of detecting a photon generated in the waveguide.

The optical spectrum is measured using a 10-nm flat-top bandpass filter (WL Photonics, tunability from 1,200 to 1,600 nm), which is set between the TEM holder and the beam splitter, connected to the optical fibres via FC/APC connectors. The transmission band was moved incrementally between 1,270 and 1,570 nm and the count rate on the detector was measured for every position (Supplementary Section 4).

The unheralded photon–photon intensity correlation is measured using superconducting nanowire single-photon detectors (Single Quantum) and a fast time tagger (Swabian Instruments, Time Tagger Ultra). The nanowire detectors show a detection efficiency of >80% and a time resolution of <20 ps, and therefore allow for sharper temporal correlation when operated in combination with a tagger unit with a root mean squared jitter of 9 ps.

Event-based electron detector

The electrons are separated in energy via a spectrometer (CEOS, CEFID) and detected on a hybrid pixel detector based on four Timepix chips with 256 × 256 readout pixels each (Amsterdam Scientific Instruments, CheeTah T3 Quad). The event-based data stream consists of electron hits (0.03-eV energy bin per pixel in the dispersion axis, 1.56-ns timing bin) and synchronized signals from two time-to-digital converters (260-ps timing bin) at which the photon detector signals arrive. This allows for temporal correlation of the electron and photon arrivals. Further post-acquisition steps are conducted to improve temporal and energy resolution on the electron side. First, a calibration of the individual pixel response times is done with regard to a mean detector response (time-of-arrival correction) to enhance time resolution²³. Second, single-particle clustering of the multiple activated pixels/hits (3.4 hits per cluster at 100-keV beam energy, maximum cluster size of 10) gives access to individual electron events. Their localization follows the arithmetic mean position and the earliest arrival time inside a single cluster. Finally, time-dependent drifts of the electron-beam position are reduced on a 10-s interval by correcting the position of the zero-loss peak maximum.

For the temporal correlation between electrons and photons, every detected electron is tagged with τ_A/B = t_A/B − t_el to the temporally closest photons on detector A and detector B (absolute propagation timing offsets are subtracted). Hence, the resulting electron–photon–photon correlation data contain the electron kinetic energy E (calibrated by the pixel position) and the (relative) arrival times of all three particles. For scanned data, the electron-beam position is also included.

A thorough analysis and quantitative fit of event rates for this three-dimensional event histogram (E, τ_A, τ_B) can be found in Supplementary Section 6. For easier access, individual cuts along or integrals over a specific axis are also analysed and fitted independently with the fit function defined by the cube fit, as described in the main text. Generally, this procedure may include heralded multi-indexing of photons within the relevant time interval of 100 ns. For calculation of the discrete heralded intensity correlation g_H⁽²⁾[q, E_el], as well as Fig. 2d, photons were assigned uniquely, only to the temporarily closest electron.

Heralding efficiency and CAR calculations

In this work, the interaction scheme is further analysed via the calculation of the intrinsic heralding efficiency and the CAR.

The intrinsic heralding efficiency quantifies how the energy-resolved electron detection predicts the number of generated photons ${\eta }_i^{I}={N}_{i,j}/({N}_j{\eta }_i^{\mathrm{d}}{T}_i)$ (ref. ⁴⁸). Here, N_i/j describes the number of detected heralding (j) or heralded (i) particles and ${\eta }_i^{\mathrm{d}}{T}_i$ is the combination of transmission losses and detector efficiencies for the heralded particle i. Rates R and counts N are generally interchangeable. In other words, the intrinsic heralding probability is the probability that, given that a heralding particle j is detected, the heralded particle i also exists. In this measurement set-up, the heralding efficiency is calculated for the case of observing m photons when selecting an energy window around a specific electron energy loss m = 1, 2 and vice versa.

The ${{\rm{CAR}}}=\frac{{R}_{{{\rm{sig}}}}-{R}_{{{\rm{acc}}}}}{{R}_{{{\rm{acc}}}}}$ (ref. ⁴⁸) relates the signal R_sig and background R_acc rates of coincidences in a specific energy loss region m. The value can be compared to a signal-to-noise ratio for correlation set-ups, analysing the quality of the coincidence process.

Heralded intensity correlation

The statistics of photons and the proof for single-photon states are often given via the intensity correlation measured in an HBT set-up, in which the signal from two single-photon detectors is cross-correlated and normalized to large time delays^45,48

$${g}^{(2)}(\tau )=\frac{{P}_{{{\rm{A}}},{{\rm{B}}}}(\tau )}{{P}_{{{\rm{A}}}}{P}_{{{\rm{B}}}}}.$$

Here, P_A,B(τ) is the probability of detector A and B being excited at a temporal delay τ from each other, and P_i (i = A, B) is the probability of detector excitation A or B.

To remove the electron influence, an electron-heralded intensity correlation is employed. The heralded intensity correlation follows⁵⁷

$${g}_{\mathrm{H}}^{(2)}({t}_{{{\rm{A}}}},{t}_{{{\rm{B}}}}| {t}_{{{\rm{e}}}})=\frac{{P}_{{{\rm{A}}},{{\rm{B}}},{{\rm{e}}}}({t}_{{{\rm{A}}}},{t}_{{{\rm{B}}}},{t}_{{{\rm{e}}}}){P}_{{{\rm{e}}}}}{{P}_{{{\rm{A}}},{{\rm{e}}}}({t}_{{{\rm{A}}}}-{t}_{{{\rm{e}}}}){P}_{{{\rm{A}}},{{\rm{e}}}}({t}_{{{\rm{B}}}}-{t}_{{{\rm{e}}}})},$$

in which the photon–photon cross-correlation is analysed in combination with the detection of a heralding particle e (see Supplementary Section 8 for more information). While the numerator P_A,B,e becomes a function of all three particle arrival times, the denominators convert into correlation functions between one photon and the heralding electron P_A/B,e as a function of τ_A/B = t_A/B − t_e. The function is again normalized to large τ_A − τ_B ≫ 0, but under the condition that either photon A or photon B is in coincidence with a herald e (τ_A/B = 0). The equation can also be written in terms of correlation counts N instead of coincidence probabilities, giving rise to equation (3).

Due to the addition of a third particle, the heralded intensity correlation is a function of τ_A and τ_B. A reduction to a single variable is done in time-averaged intensity correlation^57,58

$${\overline{g}}_{\mathrm{H}}^{(2)}({\tau }_{{{\rm{A}}}}=0,\tau ={\tau }_{{{\rm{B}}}})=\frac{{P}_{{{\rm{A}}},{{\rm{B}}},{{\rm{e}}}}(\tau ){P}_{{{\rm{e}}}}}{{P}_{{{\rm{A}}},{{\rm{e}}}}(0){P}_{{{\rm{B}}},{{\rm{e}}}}(\tau )},$$

in which one time delay τ_A is set to coincidence. Thus, the second photon B is simultaneously cross-correlated with a photon A and an electron.

Generally, all (heralded) intensity correlation functions require noise removal in the individual correlation functions, to be normalized for large time delays. However, high signal-to-noise ratios in detector counts, as experienced in this experiment, eliminate the necessity for noise removal. All shown intensity correlation measurements were conducted without background removal.

Furthermore, the heralding electrons are generally filtered to a defined energy loss to achieve photon-number-state analysis on the heralded photon side. The requirement of the electron energy loss to be a known value for photon state analysis is clarified by adding E_el as a further variable in the intensity correlation functions in equations (3) and (4).

The quality of heralding k photons via energy-filtered electrons depends on the capability of only selecting electrons at an energy loss of E₀ − kℏω. However, the broad initial electron energy distribution results in an overlap of the loss peaks, which we quantify by fitting the electron spectra (Fig. 3) and considering a specific energy selection window. The width (540 meV) and position of this window are set by optimizing the contrast (signal-to-noise ratio) of true coincidence events over the background from other peaks. The remaining electron spectral mixing reduces the CAR, as described above. The impact on the calculated g⁽²⁾(0) values depends on the observed photon number state. While the change of g⁽²⁾(0) from single-photon states due to additional heralding by zero-loss electrons is negligible, g⁽²⁾(0) of higher photon numbers may deviate from the expected values (see discussion and details for m = 1 and m = 2 in Supplementary Section 9).