Multi-photon electron emission with non-classical light

The study addresses how non-classical photon-number statistics influence non-linear photoemission from metals. Extreme-event statistics, common in systems with heavy-tailed distributions, can be engineered with quantum light such as bright squeezed vacuum (BSV) produced via high-gain parametric down-conversion. The central questions are whether the extreme, super-Poissonian photon statistics of BSV survive and are possibly enhanced in a non-linear multiphoton electron emission process, and whether the photon statistics can be imprinted onto the electron number distribution. This is important both fundamentally, entering a light–matter interaction regime beyond classical field descriptions, and for applications such as electron-beam imaging where tailored number statistics (including potential sub-Poissonian regimes) could reduce shot noise and radiation dose.

The work situates itself within research on extreme statistics in complex systems (e.g., climate and markets) and their optical analogs (optical rogue waves in nonlinear fibers). It draws on quantum optics foundations of coherent versus non-classical light statistics and properties of BSV from high-gain spontaneous parametric down-conversion, which exhibits super-bunched, heavy-tailed photon-number distributions characterized by Gamma-type laws for single-mode operation. Prior studies demonstrated that multiphoton processes (e.g., optical harmonics) inherit BSV statistics and that mode number controls fluctuation strength. In electron emission, prior event-resolved number statistics were limited to single-photon regimes or relied on electron-optical filtering with well below one electron per pulse, and Coulomb-induced energy correlations between few electrons have been observed. The present work extends these ideas to non-linear multiphoton photoemission with quantum light, testing independence of electron emission events and potential blockade effects, and examining how dispersion and multi-mode structure of BSV affect the observed statistics.

Light sources: Two pulsed sources are alternated: (i) a coherent femtosecond erbium fiber laser (central wavelength 1550 nm, 170 fs pulses); (ii) a bright squeezed vacuum (BSV) source generated via parametric down-conversion (PDC) in a 10 mm type-I BBO crystal pumped at 800 nm with ~1.6 ps pulses. BSV spectrum spans ~1400–1900 nm, centered at ~1600 nm. Both sources are pulse-picked to ~100 Hz repetition rate. Mode control of BSV: The photon-number statistics of BSV are tuned by varying the number of spatial and spectral modes. Spectral filtering: a 50 nm FWHM bandpass at 1600 nm. Spatial filtering: a slit in the back Fourier plane of the pump-focusing lens. Pump suppression is achieved by a dichroic mirror and bandpass filters. Optical delivery and target: Each beam is directed to an ultra-high vacuum chamber (~1×10^-9 hPa) and focused by an off-axis parabolic mirror to a ~5 µm (1/e^2 radius) spot onto the apex of a sharp tungsten needle tip (~10 nm radius). Assuming a field enhancement factor ~6 at the tip, coherent light reaches peak intensities up to ~5×10^12 W/cm^2 (Gaussian pulse); unfiltered BSV reaches up to ~3×10^12 W/cm^2. Electron detection: The tip is biased at -200 V; emitted electrons are accelerated to a Chevron micro-channel plate (MCP) with phosphor screen. Electron impacts create bright spots imaged by a CCD camera synchronized to the laser pulses. An image-recognition algorithm counts bright spots per frame, yielding the electron count N_e per pulse. Data sets include ~1×10^4 to 4×10^4 frames per excitation setting. The MCP background is ~0.002 counts/pulse and independent of signal. Data analysis and models: For coherent light, electron number histograms are compared to Poisson distributions with the same mean mu, and the scaling of mu versus pulse energy E_p in log-log space yields the non-linearity n (slope). For BSV, theory accounts for an n-photon process driven by a single-mode BSV with a Gamma-type photon-number distribution, leading to generalized nth-order Gamma distributions for electrons. Multi-mode BSV is modeled as m-fold convolution of single-mode distributions (statistically independent modes). Fits to experimental histograms use fixed non-linearity n determined from mu vs E_p, and vary m to best match distributions. Additional analyses consider the effects of admixture of Poissonian emission (convolution with a Poisson distribution) and the influence of dispersion-induced temporal mode mixing (group delay dispersion, GDD) on effective mode number. Additional BSV generation details (Methods): Pump pulses (1.6 ps, up to 0.4 mJ) are focused with a 700 mm cylindrical lens into the BBO crystal set near collinear degenerate type-I phase matching, with a broad beam (~3 mm) to mitigate spatial walk-off. To avoid unwanted processes (e.g., second harmonic, sum frequency, PDC saturable absorption) that could alter BSV statistics, the crystal is slightly detuned from optimal phase matching and average pump power is kept below 2 W. Residual pump is rejected by optical elements totaling OD 22 at 800 nm. Temporal resolution considerations: The number of temporal (frequency) modes detected is governed by the detector (tip) response time relative to the light coherence time. The tip emission is prompt on sub-femtosecond timescales, much shorter than the BSV coherence time with 50 nm bandwidth (~170 fs), enabling observation of single temporal mode statistics. In broadband unfiltered BSV, dispersion (GDD ~ -400 ± 100 fs^2) temporally separates photon pairs, increasing effective mode number detected.

- Coherent excitation yields Poissonian electron number statistics across a wide mean range. For pulse energies E_p = 7–13 nJ, the mean electron count mu shifts from 1 to 16. The mu vs E_p scaling gives non-linearity n = 4.4 ± 0.3, indicating predominantly four-photon photoemission. An analytical Poisson distribution with mu = 16 matches the measured histogram without fitting parameters, showing independence of emission events and no evidence for emission blockade even at large N_e. - BSV excitation with 50 nm spectral filtering shows heavy-tailed electron number distributions with maximum probability at N_e = 0 across pulse energies E_p = 9–18 nJ. The scaling of mu vs E_p gives n = 4.0 ± 0.3, consistent with four-photon excitation. - Mode-number dependence: With many spatial modes and narrow spectrum (E_p = 18 nJ, mu = 2.6), fitting with fixed n = 4 yields an effective mode number m = 11 ± 3 (multi-mode Gamma distribution). With added spatial filtering and 50 nm spectral filtering (reduced E_p up to 6 nJ), the electron distribution approaches the single-mode fourth-order distribution (m ≈ 1–2), demonstrating control over electron statistics via optical mode engineering. - Extreme events: In the near-single-mode BSV case with mu = 0.27, events up to 65 electrons per pulse are observed. Such an event under a Poisson distribution with mean 0.27 has probability ~10^-128, highlighting the extreme heavy-tail behavior imparted by BSV and the non-linear process. - Broadband, unfiltered BSV (E_p = 36 nJ) yields mu ≈ 48 and up to ~300 electrons per pulse. Due to uncompensated dispersion (GDD ~ -400 ± 100 fs^2), effective mode number increases to m = 57 ± 5, shifting the distribution peak to N_e ≈ 22 and reducing relative fluctuations, approaching Poissonian behavior as m increases. - Temporal resolution: Observation of single-mode electron statistics implies the tip’s sub-femtosecond response resolves ultrafast BSV fluctuations (coherence time ~170 fs for 50 nm bandwidth). In broadband BSV, dispersion separates correlated photon arrivals, so the prompt tip averages over more modes, explaining increased m. - Robustness: Even small admixtures (few percent) of Poissonian electron contributions would noticeably distort the single-mode Gamma distribution, but measurements remain consistent with light-statistics-dominated emission. - Transfer of statistics: Overall, the electron number distribution inherits the photon-number statistics of the driving light, and can be tuned by controlling BSV mode number.

The findings directly answer the central question: non-classical photon statistics from BSV are imprinted onto the electron number distribution in non-linear multiphoton photoemission from a metallic nanotip. The non-linearity (n ≈ 4) amplifies the heavy-tailed nature of single-mode BSV, producing electron distributions with extreme events far exceeding the mean. Conversely, increasing the number of temporal/spatial modes (via reduced coherence time or dispersion) drives the statistics toward Poissonian, demonstrating controllable tailoring of electron-beam statistics at the emission source. The Poissonian outcome under coherent excitation confirms independent emission events and absence of emission blockade even for high N_e within a pulse, despite nanometer-scale source size and femtosecond timescales. The prompt sub-femtosecond response of the tip allows it to resolve ultrafast BSV fluctuations, a capability beyond conventional photodiodes, thereby faithfully transducing optical quantum statistics into electron statistics. These results are significant for quantum electron optics and applications: Imprinting sub-Poissonian photon statistics (not demonstrated here but anticipated by symmetry of the mechanism) could reduce electron shot noise (Fano factor < 1), benefitting low-dose, high-sensitivity electron imaging, particularly of delicate biological samples. The demonstrated control over mode number and resulting statistics provides a pathway to engineered electron sources with tailored noise properties, and the high intensities used suggest opportunities in strong-field electron quantum optics with quantum light.

The work demonstrates, for the first time to the authors’ knowledge, generation of non-classical electron number statistics directly via the emission process using quantum light. Coherent femtosecond excitation yields Poissonian electron statistics consistent with independent four-photon emission. Bright squeezed vacuum excitation imprints its super-Poissonian photon statistics onto emitted electrons; by tuning spatial and spectral mode numbers, the electron statistics are tailored from near-single-mode heavy-tailed Gamma-type to multi-mode distributions approaching Poissonian. Extreme events up to 65 electrons at mean 0.27 highlight the strong amplification of fluctuations by non-linear emission. These results establish a versatile platform for transferring photon statistics to electrons (and potentially vice versa) and motivate future work to imprint sub-Poissonian statistics for shot-noise-reduced electron imaging, to explore dispersion control and mode engineering in the quantum-to-electron transduction, and to investigate strong-field quantum optics phenomena with quantum light-driven electron dynamics.

- The study demonstrates super-Poissonian (BSV) to electron statistics transfer but does not experimentally realize sub-Poissonian photon statistics; thus, claims about shot-noise-reduced electron imaging remain prospective. - Effective mode number increases in broadband BSV due to uncompensated group delay dispersion (GDD ~ -400 ± 100 fs^2), which reduces extreme-event prominence; dispersion management could further optimize statistics transfer. - Spatial and spectral filtering that approach single-mode operation also reduce available pulse energy, limiting mean electron counts and measurement dynamic range in that regime. - The analysis assumes dominance of light-driven multiphoton emission; small admixtures of Poissonian (e.g., DC field emission) contributions would distort the single-mode electron distribution, though experimental data suggest such admixtures are negligible. - Mode-number estimates rely on convolution models and fits; while consistent, direct, independent characterization of mode structure at the tip apex is not presented.