Driving electrons at needle tips strongly with quantum light

TL;DR

This paper investigates whether bright squeezed vacuum (BSV), a quantum light state with $⟨E⟩ = 0$ and intensity scaling as $I ∝ sinh^2(r)$, can drive strong-field electron dynamics at a metal needle tip. By measuring shot-resolved electron spectra and correlating each spectrum with the detected photon number $N_BSV$, the authors show that the hallmark signatures of strong-field physics—the plateau and the $10\,U_p$-cutoff—emerge only after post-selecting by $N_BSV$, consistent with projection onto a coherent-state amplitude; in contrast, shot-averaged spectra lack a plateau because BSV's large intensity fluctuations average out the structure. The authors further show that shot-averaged spectra can be reproduced as an incoherent sum of coherent-drive spectra weighted by the BSV Husimi function $Q_BSV$, explaining the presence of high-energy electrons beyond the classical cut-off. Altogether, the work links strong-field attophysics with quantum-optics formalisms, demonstrating a pseudo-coherent-state description of BSV driving and opening avenues for electron quantum-state engineering and quantum-light sensing on attosecond scales.

Abstract

Attosecond science relies on driving electrons after photoemission with the strong optical field of a laser pulse, representing an intense classical coherent state of light. Bright squeezed vacuum (BSV) is a quantum state of light intense enough to drive strong-field physics. However, its mean optical electric field is zero, suggesting that, in a semiclassical view, electrons should not experience strong driving. The question arises if and how this quantum state of light can generate attosecond science signatures in strong-field photoemission. Here we show that the key signatures of strong-field physics - the high energy plateau and the 10-$U_\mathrm{p}$-cut-off - also appear under BSV driving of a needle tip, but only when we post-select electron energy spectra on the individual photon number of each BSV pulse. When averaging over many BSV shots, we observe broad energy spectra featuring no plateau. This suggests that BSV-driven electrons behave as if driven by an ensemble of coherent states of light. Our findings bridge strong-field physics and quantum optics, offering insights into BSV and other quantum light states. Our work paves the way for electron quantum state engineering and the use of strongly driven electrons as quantum light sensors.

Figures (6)

• Figure 1: Setup for the measurement of strong-field electron energy spectra driven by quantum light.a, Bright squeezed vacuum (BSV) has a mean electric field of zero ($\langle \boldsymbol E \rangle = 0$, gray dashed line) with a large variance oscillating at twice the carrier frequency (circled inset). 96 % of the BSV are sent to the tungsten needle held in an ultrahigh vacuum chamber (not shown), while the photon number of each shot is monitored with the 4 % BSV pick-up at a photodiode (gray, losses not included). The calculated photon number distribution $P(N_\mathrm{BSV})$ for a mean of $\langle N_\mathrm{BSV} \rangle = 10^{11}$ photons per pulse (brown dashed line) is shown on the bottom left. Electrons emitted in a non-linear photoemission process are driven in the strong optical near field and rescatter at the tip surface (see boxes), generating high-energy electrons. We discuss below why the insets show classical coherent light states. b, Sketch of the home-built shot-resolving electrostatic low-energy electron spectrometer. We use an electrostatic quadrupole lens in front of the deflector to focus the electron beam emitted from the tip into the spectrometer. The energy of each electron is recorded with the help of a micro-channel plate and phosphor screen placed in the detector plane of the spectrometer (green line).
• Figure 1: Strong-field spectra measured with coherent light at a central wavelength of 1600nm. For better visibility, the spectra are shifted on the vertical axis. See text for details.
• Figure 2: Measured shot-averaged electron energy spectra.a, Electron energy spectra driven by BSV with increasing mean pulse energy (from light blue to dark blue: [$3.2,5.3,7.3,10.2,14.0, 17.0$] nJ). For better visibility, the spectra are shifted vertically (the shift of consecutive spectra is 0.4 on the logarithmic axis). The energy offset resulting from the DC bias voltage is subtracted. The red dots mark the points where the count rate has dropped to 5 % of each maximum, defining the 5%-cut-off. b, 5%-cut-off position as a function of mean pulse energy from panel a. The green dashed line is a linear fit and the blue curve is a power-law fit. The best fit exponent of the latter is $0.65\pm0.2$.
• Figure 2: Husimi function and simulated shot-averaged electron energy spectra driven by coherent light and BSV.a, Calculated amplitude part of the Husimi function $Q_\mathrm{BSV}$ (brown) and simulated electron yield (green) as function of peak intensity $I_\alpha$, shown on a double-logarithmic scale. b, Simulated CEP-averaged electron energy spectra for increasing intensities of coherent light ($[0.12, 0.54, 0.96, 1.38,1.8]\times10^{13}\,\mathrm{W/cm^2}$; light blue to dark blue). Red circles indicate the calculated cut-off positions. c, Simulated electron energy spectra for BSV with four mean intensities of $[1.2,2.5, 3.8, 5.0]\times10^{12}\,\mathrm{W/cm^2}$, equivalent to the classical cut-off positions $[3.1, 6.1, 9.0, 12.0]\,\mathrm{eV}$ (indicated by red circles). Panels b and c are identical to Fig. \ref{['fig:shot_resolved_measurement']}e, f and are shown here for clarity.
• Figure 3: Measured and simulated shot-resolved energy spectra.a, Map of the measured electron energy spectra for a fixed mean BSV pulse energy of 21 nJ. The horizontal axis shows the energy of each detected electron and the vertical axis the BSV photon number $N_\mathrm{BSV}$ measured at the photodiode (Fig. \ref{['fig:setup']}a). For increasing $N_\mathrm{BSV}$, the energy spectra broaden substantially, from close to the minimum width of 2 eV up to the maximum detectable electron energy of 65 eV. b, Six line-out spectra from a. The positions of the lineouts are indicated by blue arrows in a. The shape of these lineout spectra resembles that of well-known strong-driving spectra - with classical coherent laser light: They clearly show the plateau and the 10-$U_\mathrm{p}$-cut-off (red dots). c, Cut-off positions from all line-outs in a as a function of the photon number $N_\mathrm{BSV}$. The right-hand axis indicates the expected intensity $I_\mathrm{clas}$ for coherent light calculated back from the 10-$U_\mathrm{p}$-law (see text). The red dashed line is a linear fit to the data, indicating a scaling behavior identical with coherent driving. d, Simulated shot-resolved electron energy spectra from integrating the time-dependent Schrödinger equation. The vertical axis shows the intensity of a coherent driver $I_\alpha$. The shape matches the experimentally obtained one well, and we stress that these numerical results are obtained with a classical driving field. The fast oscillations are a result of inter- and intra-cycle effects Arb2010Kruger2011, not visible in the experiment. We note further that we find no electrons at negative energies because the simulation is single electron-based (see Methods). Extended Data Fig. \ref{['fig:Methods_shot_resolved_measurement']} contains a version of this figure with more detailed and additional discussions. e, Line-out energy spectra from d for increasing intensities of coherent light ($[0.12, 0.54, 0.96, 1.38,1.8]\times10^{13}\,\mathrm{W/cm^2}$; light blue to dark blue and indicated by blue arrows in d). Red circles indicate the calculated cut-off positions. f, Simulated electron energy spectra for BSV with four mean intensities of $[1.2,2.5, 3.8, 5.0]\times10^{12}\,\mathrm{W/cm^2}$, equivalent to the classical cut-off positions $[3.1, 6.1, 9.0, 12.0]\,\mathrm{eV}$ (indicated by red circles).