Electron-Cathodoluminescence Photon Pairs

• Electron-cathodoluminescence photon pairs are produced when fast electrons excite optical modes, linking electron energy loss to the timing of emitted photons.
• They enable precise analysis of excitation pathways and photon statistics, distinguishing classical burst behavior from potential quantum correlations.
• Advanced coincidence spectroscopy techniques, such as CLE and HBT interferometry, extract lifetime and kinematic information with high temporal resolution.

Electron-cathodoluminescence photon pairs arise when a fast electron in an electron microscope generates optical emission and the resulting quanta are treated in coincidence. In current usage, the expression covers two distinct but related observables: hybrid electron–photon pairs, in which a detected cathodoluminescence (CL) photon is matched to the exciting electron, and photon–photon pairs, in which two photons are emitted within the same electron-triggered burst and are quantified through second-order intensity correlations. The first observable underlies cathodoluminescence excitation spectroscopy (CLE), time-correlated electron and photon counting microscopy, single-photon recoil measurements, ghost imaging, and continuous-variable entanglement tests; the second underlies superbunching studies in incoherent CL from defects, excitons, and quantum wells (Varkentina et al., 2023, Preimesberger et al., 17 Apr 2025, Fiedler et al., 2021).

1. Conceptual scope and physical regimes

Electron-driven CL separates naturally into coherent and incoherent regimes. In incoherent CL, the electron excites material states that later radiatively decay stochastically, producing lifetime-limited luminescence that carries spectral fingerprints of transitions but no definite phase relation to the driving electron. In coherent CL, the electron directly drives electromagnetic modes that radiate promptly and coherently within the electron’s interaction time; the emitted photon is then kinematically tied to the electron’s energy loss and recoil. Transition radiation, Cherenkov radiation, Smith–Purcell radiation, and emission from localized modes such as Mie or plasmonic resonances are the canonical coherent channels (Preimesberger et al., 2024, Yanagimoto et al., 2023).

This distinction is decisive for the meaning of a “pair.” In electron–photon coincidence spectroscopy, each pair links one electron timestamp and, when available, one electron energy or momentum measurement to one photon timestamp. In photon-statistics experiments, by contrast, “pairs” usually denote two photons detected within the same electron-excitation burst. The two observables are not interchangeable. Electron–photon pairs probe excitation-to-emission pathways, recoil, and joint kinematics, whereas photon–photon pairs probe burst statistics, emitter multiplicity, and cascade dynamics (Varkentina et al., 2023, Yuge et al., 2022).

A recurring misconception is that any pronounced zero-delay coincidence peak implies nonclassicality. The CL literature distinguishes this sharply. Large photon bunching, including giant or “superbunched” values of $g^{(2)}(0)$, can arise from classical intensity intermittency or from multi-step cascades in incoherent CL. Conversely, coherent CL can support electron–photon entanglement while retaining Poisson per-event photon-number statistics. A large $g^{(2)}(0)$ alone therefore does not imply entanglement, and prompt electron–photon coincidence alone does not distinguish classical correlation from certified nonseparability (Yanagimoto et al., 2024, Yuge et al., 2022).

2. Coincidence spectroscopy and lifetime extraction

The most developed electron–photon pairing methods time-tag individual electrons and photons and then construct joint observables. In CLE, each transmitted electron is recorded by an event-based electron energy-loss spectroscopy (EELS) detector with its energy loss $E$ and arrival time $t_e$, while each emitted photon is recorded by a single-photon detector with arrival time $t_\gamma$. A search algorithm builds two-dimensional histograms $H(E,\Delta t)$ with $\Delta t=t_e-t_\gamma$, so that correlated electron–photon pairs accumulate near $\Delta t\approx 0$, whereas accidental coincidences produce a flat background at larger delays. The corresponding cross-correlation is

$$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$

For a single-exponential decay, the ideal delay distribution is

$$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$

and the measured histogram is modeled as

$g^{(2)}(0)$0

with $g^{(2)}(0)$1 the coincidence amplitude, $g^{(2)}(0)$2 the accidental background, and the instrumental response function (IRF) often approximated by a Gaussian (Varkentina et al., 2023).

A central experimental result of CLE is that transition radiation (TR), whose intrinsic duration is much shorter than the measured response, serves as a prompt reference for characterizing the IRF. In nanodiamonds with $g^{(2)}(0)$3 emission and in h-BN with a $g^{(2)}(0)$4 eV defect emission, the IRF was found to be typically $g^{(2)}(0)$5 ns and, under optimal conditions, $g^{(2)}(0)$6 ns. Against that response, measured lifetimes of $g^{(2)}(0)$7 centers in diamond nanoparticles were $g^{(2)}(0)$8 to $g^{(2)}(0)$9 ns across eighteen independent measurements, whereas the h-BN $E$0 eV defect remained barely distinguishable from the IRF, i.e. $E$1 ns. Because CLE bins coincidences by electron energy loss, it also accesses $E$2 and separates prompt TR at low losses from longer-lived defect luminescence at higher losses (Varkentina et al., 2023).

Time-correlated electron and photon counting microscopy generalizes the coincidence idea to electron-triggered lifetime metrology that does not require a pulsed beam. In that framework, one detector receives the sample photon signal and another receives scintillator light proportional to transmitted electron arrivals, producing an asymmetric cross-correlation

$E$3

where $E$4 is the sample lifetime, $E$5 the scintillator decay time, and $E$6 an excitation correlation factor. The $E$7 branch depends only on $E$8, while $E$9 depends only on $t_e$0, so the emitter lifetime is obtained independently of scintillator decay. This was demonstrated for $t_e$1 nm nanodiamond particles with $t_e$2 NV centers per particle, for coherent CL from Au nanoparticles, and in a simplified geometry where a $t_e$3 perovskite acted as a built-in scintillator substrate (Yanagimoto et al., 2023).

3. Photon-statistics theory and the origin of bunching

Photon–photon pairs in CL are usually characterized through Hanbury Brown–Twiss interferometry and the normalized second-order correlation function

$t_e$4

For CL generated by multi-emitter bursts, the theoretical problem is to connect the electron-triggered excitation cascade to the zero-delay excess and to the decay of $t_e$5 back to unity. A master-equation treatment with $t_e$6 identical, noninteracting two-level systems excited simultaneously by an incoming electron yields the exact zero-delay value

$t_e$7

where $t_e$8 is the $t_e$9th harmonic number, $t_\gamma$0 the radiative lifetime, and $t_\gamma$1 the electron-induced excitation rate. In the low-excitation regime, this reduces to a large $t_\gamma$2 that scales approximately as $t_\gamma$3, and the time dependence is well approximated by

$t_\gamma$4

Within this model, superbunching results from a mixture of an excited photon state and the vacuum state, rather than from a coherent two-photon state (Yuge et al., 2022).

A complementary event-conditioned formulation asks how many photons a single electron generates. If $t_\gamma$5 is the probability that one electron-excitation event emits $t_\gamma$6 photons, then the intrinsic per-event correlation factor is

$t_\gamma$7

The measured bunching becomes

$t_\gamma$8

with $t_\gamma$9 the electron current and $H(E,\Delta t)$0 the correlation-time width. In this picture, coherent CL corresponds to per-event Poisson statistics with $H(E,\Delta t)$1, while incoherent CL involving mediator cascades becomes super-Poissonian with $H(E,\Delta t)$2 (Yanagimoto et al., 2024).

For incoherent CL under continuous and pulsed electron beams, the same logic can be made fully analytical. In InGaN/GaN quantum wells, an incident electron first excites bulk plasmons; these decay into hot carriers, which then feed the radiative channel. The electron excitation efficiency is written as $H(E,\Delta t)$3, where $H(E,\Delta t)$4 is the mean number of bulk plasmons excited per electron, and in the ultrashort-pulse limit the bunching simplifies to

$H(E,\Delta t)$5

with $H(E,\Delta t)$6 the average number of electrons per pulse. This formalism explains why $H(E,\Delta t)$7 decreases as the number of electrons per pulse grows, even when the pulse duration is only a few picoseconds (Solà-Garcia et al., 2020).

4. Representative material systems and experimental phenomenology

The experimental landscape of CL photon pairs is dominated by defect and exciton systems in which a single electron can synchronize multiple radiative emitters. In nanodiamonds containing $H(E,\Delta t)$8 centers, room-temperature STEM excitation at $H(E,\Delta t)$9 keV produced $\Delta t=t_e-t_\gamma$0 across $\Delta t=t_e-t_\gamma$1 measurements, with a mean fitted effective lifetime $\Delta t=t_e-t_\gamma$2 ns. Spectrally resolved HBT showed that the bunching is mediated by the $\Delta t=t_e-t_\gamma$3 phonon sidebands, while no observable bunching is detected at the zero-phonon line. Plasmonic mediation on a silver nanoplate increased CL brightness by a factor of $\Delta t=t_e-t_\gamma$4 across the spectrum but decreased $\Delta t=t_e-t_\gamma$5 by roughly $\Delta t=t_e-t_\gamma$6, consistent with single surface plasmon polaritons exciting single, temporally uncorrelated NV centers (Feldman et al., 2017).

A distinct nanodiamond regime appears under indirect electron excitation. For ensembles of $\Delta t=t_e-t_\gamma$7 centers in $\Delta t=t_e-t_\gamma$8 nm nanodiamonds, $\Delta t=t_e-t_\gamma$9 remained single-exponential with $\Delta t\approx 0$0 ns on Al and $\Delta t\approx 0$1 ns on $\Delta t\approx 0$2, but $\Delta t\approx 0$3 increased exponentially with beam distance from the nanodiamond. In the maps shown, $\Delta t\approx 0$4 reached values as high as $\Delta t\approx 0$5 under indirect excitation at tens to hundreds of nanometers from the emitter, while the abstract reports values approaching $\Delta t\approx 0$6. The interpretation is that rare, intense bursts driven by secondary electrons, substrate-supported plasmons, or aloof near-field excitation increase the variance of photon number relative to its mean (Iyer et al., 2023).

In two-dimensional semiconductors, hBN-encapsulated $\Delta t\approx 0$7 monolayers have yielded the most extreme superbunching values reported in the supplied corpus. Under electron-beam excitation, the neutral exciton dominates the CL spectrum near $\Delta t\approx 0$8 nm, and the bunching peak at $\Delta t\approx 0$9 reflects nearly synchronous recombination of many identical excitons excited by a single electron. The measured values were $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$0 at $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$1 pA and $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$2 kV for the bare $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$3–hBN heterostructure, $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$4 at $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$5 pA, and $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$6 at $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$7 kV with $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$8 pA. Adding a thin monocrystalline gold nanodisk increased the bunching to $$C_{e\gamma}(\Delta t)=\langle I_e(t)I_\gamma(t+\Delta t)\rangle .$$9 at $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$0 pA and $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$1 kV and to a record-high $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$2 at $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$3 pA and $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$4 kV. The extracted effective radiative lifetimes were $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$5 ps, $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$6 ps, and $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$7 ps. The paper explicitly states that this superbunching does not arise from biexciton cascade dynamics; it is classical synchronization of many identical excitons by single electrons (Fiedler et al., 2021).

Quantum-well CL under continuous, nanosecond-pulsed, and ultrashort picosecond-pulsed beams illustrates the same physics in a form amenable to analytical inversion. In InGaN/GaN quantum wells, the reported continuous-beam value was $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$8 at $$p(\Delta t)=\Theta(\Delta t)\,\frac{1}{\tau}e^{-\Delta t/\tau},$$9 pA and $g^{(2)}(0)$00 keV. Fitting yielded excitation efficiencies of $g^{(2)}(0)$01 and $g^{(2)}(0)$02 for $g^{(2)}(0)$03 and $g^{(2)}(0)$04 keV electron beams, respectively, and the ultrashort-pulse experiments extended to up to $g^{(2)}(0)$05 electrons per pulse without compelling evidence that non-linear effects play a significant role. These measurements established that the relevant “pairs” remain classical bunching events associated with a single electron’s excitation cascade and the emitter lifetime, rather than nonclassical two-photon states (Solà-Garcia et al., 2020).

5. Coherent cathodoluminescence, single-photon recoil, and entanglement

In coherent CL, the pair observable changes from burst statistics to single-event kinematics. When a fast electron emits a coherent CL photon, the event obeys

$g^{(2)}(0)$06

so that for small deflections the transverse recoil gives $g^{(2)}(0)$07 and the measured electron energy loss satisfies $g^{(2)}(0)$08. In a $g^{(2)}(0)$09 keV TEM experiment on a $g^{(2)}(0)$10 nm-thick monocrystalline silicon membrane, time-tagged electron–photon coincidences with a window of $g^{(2)}(0)$11 ns isolated the weak radiative signal from dominant non-radiative backgrounds. In the $g^{(2)}(0)$12–$g^{(2)}(0)$13 eV electron energy-loss window, coincidence filtering enhanced the transition-radiation signal by a factor $g^{(2)}(0)$14 compared to energy filtering alone; the measured coincidence-to-accidental ratio was $g^{(2)}(0)$15, the observed cross-correlation peak had FWHM $g^{(2)}(0)$16 ns, and the overall electron–photon coincidence timing resolution was limited to $g^{(2)}(0)$17 ns (Preimesberger et al., 2024).

The same hybrid pairs also support coincidence imaging. In a later TEM experiment on a $g^{(2)}(0)$18 nm monocrystalline Si membrane, near-field and far-field ghost images were reconstructed by correlating time-tagged electrons with photons transmitted through periodic masks. Using the Mancini–Giovannetti–Vitali–Tombesi criterion with $g^{(2)}(0)$19 and $g^{(2)}(0)$20, the measured bounds were $g^{(2)}(0)$21 and $g^{(2)}(0)$22, giving

$g^{(2)}(0)$23

which violated the separable-state bound by approximately $g^{(2)}(0)$24 standard deviations. Mixed conjugate bases showed no ghost images, providing the control expected for non-commuting observables (Preimesberger et al., 17 Apr 2025).

A related free-space CL implementation used a custom parabolic mirror and a photon-arm mask to image complex patterns with electron–photon coincidences only. In that experiment, more than $g^{(2)}(0)$25 coincidences were recorded over an uninterrupted $g^{(2)}(0)$26-hour acquisition, and the measured resolution at the TEM sample plane was $g^{(2)}(0)$27, corresponding to $g^{(2)}(0)$28. The paper does not claim entanglement; it attributes the imaging to strong time, energy, and position correlations produced predominantly by transition radiation and isolated by energy filtering and temporal gating (Bogdanov et al., 18 Sep 2025).

The most general theoretical account of coherent CL pairs in the supplied literature reconstructs the scattered electron–photon state directly from the luminescence spectrum $g^{(2)}(0)$29:

$g^{(2)}(0)$30

From this state one can compute the subsystem purity $g^{(2)}(0)$31, the Schmidt number $g^{(2)}(0)$32, and an EPR-type witness $g^{(2)}(0)$33. The theory distinguishes wave-like entangled, particle-like entangled, and classical regimes and identifies the roles of transverse electron coherence, longitudinal electron coherence, and photon spectral width in determining whether strong spatial entanglement emerges (Yuge et al., 22 May 2026).

6. Technical limits, interpretation, and prospective directions

Across the field, the dominant limitations are timing resolution, collection efficiency, and background discrimination. In CLE, the IRF is typically about $g^{(2)}(0)$34 ns and is dominated by event-based EELS timing and multimode-fiber dispersion, with zero-delay shifts of $g^{(2)}(0)$35–$g^{(2)}(0)$36 ns. In coherent TEM coincidence work, the electron detector jitter and synchronization presently limit the electron–photon timing resolution to about $g^{(2)}(0)$37 ns. Additional sources of bias include detector dark counts, ambient photons, cosmic rays, dead time, pileup, spectrometer drift, mirror shading, horseshoe-shaped angular acceptance, fiber transmission loss, and contamination-induced spurious CL (Varkentina et al., 2023, Preimesberger et al., 2024).

Interpretively, the literature has converged on several points. First, electron–photon pairs and photon–photon pairs are complementary rather than competing observables. Electron–photon coincidence directly connects excitation to emission and exposes energy–momentum conservation, recoil, and entanglement; photon–photon coincidence diagnoses the statistics of the radiative burst generated by each electron. Second, superbunching is not in itself a signature of a nonclassical source. The master-equation and cascade models explicitly interpret large $g^{(2)}(0)$38 as arising from a sparse train of bright bursts mixed with vacuum or from super-Poissonian mediator cascades. Third, coherent CL can support nonclassical electron–photon correlations even when the photon-number statistics per event remain Poissonian (Yuge et al., 2022, Yanagimoto et al., 2024).

The most immediate technical route forward is improved hardware. The CLE work identifies Timepix4-class timing and monomode or free-space photon collection as routes to sub-nanosecond resolution. The recoil and entanglement work points to higher-efficiency, lower-jitter photon detectors such as SNSPDs, finer angular resolution, and phase-stable electron interferometry. These changes would reduce accidentals, strengthen heralding, and permit more stringent quantitative joint-state reconstruction (Varkentina et al., 2023, Preimesberger et al., 2024).

A separate direction is optical-mode engineering. Silica microspheres have been analyzed as CL emitters and collimating resonators whose photon generation occurs at the sphere surface and whose emitted beam can occupy a solid angle $g^{(2)}(0)$39 sr, corresponding to $g^{(2)}(0)$40 in air, with $g^{(2)}(0)$41, FWHM $g^{(2)}(0)$42, emittance $g^{(2)}(0)$43, and beam quality $g^{(2)}(0)$44. The paper argues that such collimated and directed CL in free space will enhance existing quantum measurements of CL and facilitate new ones, such as high-rate electron–photon entangled pairs, CL from quantum emitters, and homodyne analysis of CL (Aharon et al., 6 Nov 2025).

Taken together, these developments define an emerging hierarchy of CL pair physics. At one level, a single electron excites many emitters and produces classically correlated photon bursts with enormous $g^{(2)}(0)$45. At another, the same electron can be paired event-by-event with a single coherent CL photon, revealing recoil, dispersion, ghost images, and continuous-variable entanglement. The unifying thread is coincidence: once the electron and photon are recorded as a joint event rather than as independent averages, cathodoluminescence becomes a spectroscopy of pathways, kinematics, and correlations rather than a spectroscopy of light intensity alone.