Ponderomotive Generation and Detection of Attosecond Free-Electron Pulse Trains

Quantum effects in the interaction of low-energy electrons with light

The interaction between free electrons and optical fields constitutes a unique platform to investigate ultrafast processes in matter and explore fundamental quantum phenomena. Specifically, optically modulated electrons in ultrafast electron microscopy act as noninvasive probes that push space-time-energy resolution to the picometer-attosecond-microelectronvolt range. Electron energies well above the involved photon energies are commonly used, rendering a low electron-light coupling and, thus, only providing limited access to the wealth of quantum nonlinear phenomena underlying the dynamical response of nanostructures. Here, we theoretically investigate electron-light interactions between photons and electrons of comparable energies, revealing quantum and recoil effects that include a nonvanishing coupling of surface-scattered electrons to light plane waves, inelastic electron backscattering from confined optical fields, and strong electron-light coupling under grazing electron diffraction by an illuminated crystal surface. Our exploration of electron-light-matter interactions holds potential for applications in ultrafast electron microscopy.

Field-Induced Rocking-Curve Effects in Attosecond Electron Diffraction

Recent advances in electron microscopy trigger the question of whether attosecond electron diffraction can resolve atomic-scale electron dynamics in crystalline materials in space and time. Here, we explore the ultrafast dynamics of the relevant electron-lattice scattering process. We drive a single-crystalline silicon membrane with the optical cycles of near-infrared laser light and use phase-locked attosecond electron pulses to produce electron diffraction patterns as a function of delay. For all Bragg spots, we observe time-dependent intensity changes and position shifts that are correlated with a time shift of 0.5-1.2  fs. For single-cycle excitation pulses with strong peak intensity, the correlations become nonlinear. The origins of these effects are local and integrated beam deflections by the optical electric and magnetic fields at the crystal membrane. Those deflections modify the diffraction intensities in addition to the atomic structure factor dynamics by time-dependent rocking-curve effects. However, the measured time delays and symmetries allow one to disentangle both effects. Future attosecond electron diffraction and microscopy experiments need to be based on these results.

Strongly correlated multielectron bunches from interaction with quantum light

Strongly correlated electron systems are a cornerstone of modern physics, being responsible for groundbreaking phenomena from superconducting magnets to quantum computing. In most cases, correlations in electrons arise exclusively because of Coulomb interactions. In this work, we reveal that free electrons interacting simultaneously with a light field can become highly correlated via mechanisms beyond Coulomb interactions. In the case of two electrons, the resulting Pearson correlation coefficient for the joint probability distribution of the output electron energies is enhanced by more than 13 orders of magnitude compared to that of electrons interacting with the light field in succession (one after another). These highly correlated electrons are the result of momentum and energy exchange between the participating electrons via the external quantum light field. Our findings pave the way to the creation and control of highly correlated free electrons for applications including quantum information and ultrafast imaging.

Hollow metal tubes for efficient electron manipulation using terahertz surface waves

Compact electron sources have been instrumental in multidiscipline sciences including fundamental physics, oncology treatments, and advanced industries. Of particular interest is the terahertz-driven electron manipulation that holds great promise for an efficient high gradient of multi-GeV/m inside a regular dielectric-lined waveguide (DLW). The recent study relying on terahertz surface waves has demonstrated both high terahertz energy and improved coupling efficiency with the DLW. However, the large energy spread pertaining to the laser-induced electron pulse impedes the practical use of the system. Here, we propose a scheme for extending the idea of surface-wave-driven electron manipulation to mature electron sources such as commercial direct-current and radio-frequency electron guns. By using a simple hollow cylinder tube for electron transmission, we show that the electron energy modulation can reach up to 860 keV, or compress the electron pulse width to 15 fs using a 2.9 mJ single-cycle terahertz pulse. The trafficability of the hollow tube also allows for a cascade of the system, which is expected to pave the way for compact and highly efficient THz-driven electron sources.

Multicolor x-rays from free electron-driven van der Waals heterostructures

The emergence of van der Waals (vdW) heterostructures has led to precise and versatile methods of fabricating devices with atomic-scale accuracies. Hence, vdW heterostructures have shown much promise for technologies including photodetectors, photocatalysis, photovoltaic devices, ultrafast photonic devices, and field-effect transistors. These applications, however, remain confined to optical and suboptical regimes. Here, we theoretically show and experimentally demonstrate the use of vdW heterostructures as platforms for multicolor x-ray generation. By driving the vdW heterostructures with free electrons in a table-top setup, we generate x-ray photons whose output spectral profile can be user-customized via the heterostructure design and even controlled in real time. We show that the multicolor photon energies and their corresponding intensities can be tailored by varying the electron energy, the electron beam position, as well as the geometry and composition of the vdW heterostructure. Our results reveal the promise of vdW heterostructures in realizing highly versatile x-ray sources for emerging applications in advanced x-ray imaging and spectroscopy.

Accelerating Quantum Materials Development with Advances in Transmission Electron Microscopy

Quantum materials are driving a technology revolution in sensing, communication, and computing, while simultaneously testing many core theories of the past century. Materials such as topological insulators, complex oxides, superconductors, quantum dots, color center-hosting semiconductors, and other types of strongly correlated materials can exhibit exotic properties such as edge conductivity, multiferroicity, magnetoresistance, superconductivity, single photon emission, and optical-spin locking. These emergent properties arise and depend strongly on the material's detailed atomic-scale structure, including atomic defects, dopants, and lattice stacking. In this review, we describe how progress in the field of electron microscopy (EM), including in situ and in operando EM, can accelerate advances in quantum materials and quantum excitations. We begin by describing fundamental EM principles and operation modes. We then discuss various EM methods such as (i) EM spectroscopies, including electron energy loss spectroscopy (EELS), cathodoluminescence (CL), and electron energy gain spectroscopy (EEGS); (ii) four-dimensional scanning transmission electron microscopy (4D-STEM); (iii) dynamic and ultrafast EM (UEM); (iv) complementary ultrafast spectroscopies (UED, XFEL); and (v) atomic electron tomography (AET). We describe how these methods could inform structure-function relations in quantum materials down to the picometer scale and femtosecond time resolution, and how they enable precision positioning of atomic defects and high-resolution manipulation of quantum materials. For each method, we also describe existing limitations to solve open quantum mechanical questions, and how they might be addressed to accelerate progress. Among numerous notable results, our review highlights how EM is enabling identification of the 3D structure of quantum defects; measuring reversible and metastable dynamics of quantum excitations; mapping exciton states and single photon emission; measuring nanoscale thermal transport and coupled excitation dynamics; and measuring the internal electric field and charge density distribution of quantum heterointerfaces- all at the quantum materials' intrinsic atomic and near atomic-length scale. We conclude by describing open challenges for the future, including achieving stable sample holders for ultralow temperature (below 10K) atomic-scale spatial resolution, stable spectrometers that enable meV energy resolution, and high-resolution, dynamic mapping of magnetic and spin fields. With atomic manipulation and ultrafast characterization enabled by EM, quantum materials will be poised to integrate into many of the sustainable and energy-efficient technologies needed for the 21st century.

Weak measurements and quantum-to-classical transitions in free electron-photon interactions

How does the quantum-to-classical transition of measurement occur? This question is vital for both foundations and applications of quantum mechanics. Here, we develop a new measurement-based framework for characterizing the classical and quantum free electron-photon interactions and then experimentally test it. We first analyze the transition from projective to weak measurement in generic light-matter interactions and show that any classical electron-laser-beam interaction can be represented as an outcome of weak measurement. In particular, the appearance of classical point-particle acceleration is an example of an amplified weak value resulting from weak measurement. A universal factor, [Formula: see text], quantifies the measurement regimes and their transition from quantum to classical, where [Formula: see text] corresponds to the ratio between the electron wavepacket size and the optical wavelength. This measurement-based formulation is experimentally verified in both limits of photon-induced near-field electron microscopy and the classical acceleration regime using a DLA. Our results shed new light on the transition from quantum to classical electrodynamics, enabling us to employ the essence of the wave-particle duality of both light and electrons in quantum measurement for exploring and applying many quantum and classical light-matter interactions.

Sub-terahertz nearfields for electron-pulse compression

The advent of ultrafast science with pulsed electron beams raised the need to control the temporal features of the electron pulses. One promising suggestion is the nano-selective quantum optics with multi-electrons, which scales quadratically with the number of electrons within the coherence time of the quantum system. Terahertz (THz) radiation from optical nonlinear crystals is an attractive methodology to generate the rapidly varying electric fields necessary for electron compression, with the advantage of an inherent temporal locking to laser-triggered electrons, such as in ultrafast electron microscopes. Longer (picosecond-) pulses require a sub-THz field for their compression. However, the generation of such low frequencies requires pumping with energetic optical pulses and their focusability is fundamentally limited by their mm-wavelength. This work proposes electron-pulse compression with sub-THz fields directly in the vicinity of their dipolar origin, thereby avoiding mediation through radiation. We analyze the merits of nearfields for compression of slow electrons, particularly in challenging regimes for THz radiation, such as small numerical apertures, micro-joule-level optical pump pulses, and low frequencies. This scheme can be implemented within the tight constraints of electron microscopes and reach fields of a few kV/cm below 0.1 THz at high repetition rates. Our paradigm offers a realistic approach for controlling electron pulses spatially and temporally in many experiments, opening the path of flexible multi-electron manipulation for analytic and quantum sciences.

Lorentz microscopy of optical fields

In electron microscopy, detailed insights into nanoscale optical properties of materials are gained by spontaneous inelastic scattering leading to electron-energy loss and cathodoluminescence. Stimulated scattering in the presence of external sample excitation allows for mode- and polarization-selective photon-induced near-field electron microscopy (PINEM). This process imprints a spatial phase profile inherited from the optical fields onto the wave function of the probing electrons. Here, we introduce Lorentz-PINEM for the full-field, non-invasive imaging of complex optical near fields at high spatial resolution. We use energy-filtered defocus phase-contrast imaging and iterative phase retrieval to reconstruct the phase distribution of interfering surface-bound modes on a plasmonic nanotip. Our approach is universally applicable to retrieve the spatially varying phase of nanoscale fields and topological modes.

Inelastic electron scattering at a single-beam structured light wave

In free space, electrons undergo inelastic scattering in the presence of ponderomotive potentials generated by light pulses and standing light waves. The resulting modulated electron energy spectrum can exhibit the formation of discrete energy sidebands when multiple light beams are employed. Here, we demonstrate the inelastic scattering of slow-electron wavepackets at a propagating Hermite-Gaussian light beam. The pulsed Hermite-Gaussian beam thus forms a ponderomotive potential for the electron with sufficient momentum components, leading to the inelastic scattering and subsequent formation of discrete energy sidebands. We show that the resulting energy-gain spectra after the interaction are strongly influenced by the self-interference of the electrons in this ponderomotive potential. This effect is observable across various wavelengths, and the energy modulation can be controlled by varying the electron velocity and light intensity. By utilizing the vast landscape of structured electromagnetic fields, this effect introduces an additional platform for manipulating electron wavepackets.

Spatiotemporal Electron Beam Focusing through Parallel Interactions with Shaped Optical Fields

The ability to modulate free electrons with light has emerged as a powerful tool to produce attosecond electron wave packets. However, research has so far aimed at the manipulation of the longitudinal wave function component, while the transverse degrees of freedom have primarily been utilized for spatial rather than temporal shaping. Here, we show that the coherent superposition of parallel light-electron interactions in transversally separate zones allows for a simultaneous spatial and temporal compression of a convergent electron wave function, enabling the formation of sub-Ångström focal spots of attosecond duration. Specifically, spots spanning just ∼3% of the light optical cycle are shown to be formed, accompanied by an increase by only a factor of 2 in spatial extension relative to an unperturbed beam. The proposed approach will facilitate the exploration of previously inaccessible ultrafast atomic-scale phenomena, in particular enabling attosecond scanning transmission electron microscopy.

Single-Pixel Imaging in Space and Time with Optically Modulated Free Electrons

Single-pixel imaging, originally developed in light optics, facilitates fast three-dimensional sample reconstruction as well as probing with light wavelengths undetectable by conventional multi-pixel detectors. However, the spatial resolution of optics-based single-pixel microscopy is limited by diffraction to hundreds of nanometers. Here, we propose an implementation of single-pixel imaging relying on attainable modifications of currently available ultrafast electron microscopes in which optically modulated electrons are used instead of photons to achieve subnanometer spatially and temporally resolved single-pixel imaging. We simulate electron beam profiles generated by interaction with the optical field produced by an externally programmable spatial light modulator and demonstrate the feasibility of the method by showing that the sample image and its temporal evolution can be reconstructed using realistic imperfect illumination patterns. Electron single-pixel imaging holds strong potential for application in low-dose probing of beam-sensitive biological and molecular samples, including rapid screening during in situ experiments.

Quantum Interference between Fundamentally Different Processes Is Enabled by Shaped Input Wavefunctions

This work presents a general framework for quantum interference between processes that can involve different fundamental particles or quasi-particles. This framework shows that shaping input wavefunctions is a versatile and powerful tool for producing and controlling quantum interference between distinguishable pathways, beyond previously explored quantum interference between indistinguishable pathways. Two examples of quantum interference enabled by shaping in interactions between free electrons, bound electrons, and photons are presented: i) the vanishing of the zero-loss peak by destructive quantum interference when a shaped electron wavepacket couples to light, under conditions where the electron's zero-loss peak otherwise dominates; ii) quantum interference between free electron and atomic (bound electron) spontaneous emission processes, which can be significant even when the free electron and atom are far apart, breaking the common notion that a free electron and an atom must be close by to significantly affect each other's processes. Conclusions show that emerging quantum wave-shaping techniques unlock the door to greater versatility in light-matter interactions and other quantum processes in general.

Tunable photon-induced spatial modulation of free electrons

Spatial modulation of electron beams is an essential tool for various applications such as nanolithography and imaging, yet its conventional implementations are severely limited and inherently non-tunable. Conversely, proposals of light-driven electron spatial modulation promise tunable electron wavefront shaping, for example, using the mechanism of photon-induced near-field electron microscopy. Here we present tunable photon-induced spatial modulation of electrons through their interaction with externally controlled surface plasmon polaritons (SPPs). Using recently developed methods of shaping SPP patterns, we demonstrate a dynamic control of the electron beam with a variety of electron distributions and verify their coherence through electron diffraction. Finally, the nonlinearity stemming from energy post-selection provides us with another avenue for controlling the electron shape, generating electron features far below the SPP wavelength. Our work paves the way to on-demand electron wavefront shaping at ultrafast timescales, with prospects for aberration correction, nanofabrication and material characterization.

Ultrafast Electron Microscopy of Nanoscale Charge Dynamics in Semiconductors

The ultrafast dynamics of charge carriers in solids plays a pivotal role in emerging optoelectronics, photonics, energy harvesting, and quantum technology applications. However, the investigation and direct visualization of such nonequilibrium phenomena remains as a long-standing challenge, owing to the nanometer-femtosecond spatiotemporal scales at which the charge carriers evolve. Here, we propose and demonstrate an interaction mechanism enabling nanoscale imaging of the femtosecond dynamics of charge carriers in solids. This imaging modality, which we name charge dynamics electron microscopy (CDEM), exploits the strong interaction of free-electron pulses with terahertz (THz) near fields produced by the moving charges in an ultrafast scanning transmission electron microscope. The measured free-electron energy at different spatiotemporal coordinates allows us to directly retrieve the THz near-field amplitude and phase, from which we reconstruct movies of the generated charges by comparison to microscopic theory. The CDEM technique thus allows us to investigate previously inaccessible spatiotemporal regimes of charge dynamics in solids, providing insight into the photo-Dember effect and showing oscillations of photogenerated electron-hole distributions inside a semiconductor. Our work facilitates the exploration of a wide range of previously inaccessible charge-transport phenomena in condensed matter using ultrafast electron microscopy.

Attosecond electron-beam technology: a review of recent progress

Electron microscopy and diffraction with ultrashort pulsed electron beams are capable of imaging transient phenomena with the combined ultrafast temporal and atomic-scale spatial resolutions. The emerging field of optical electron beam control allowed the manipulation of relativistic and sub-relativistic electron beams at the level of optical cycles. Specifically, it enabled the generation of electron beams in the form of attosecond pulse trains and individual attosecond pulses. In this review, we describe the basics of the attosecond electron beam control and overview the recent experimental progress. High-energy electron pulses of attosecond sub-optical cycle duration open up novel opportunities for space-time-resolved imaging of ultrafast chemical and physical processes, coherent photon generation, free electron quantum optics, electron-atom scattering with shaped wave packets and laser-driven particle acceleration. Graphical Abstract.

Optical manipulation of matter waves

Light is used to steer the motion of atoms in free space, enabling cooling and trapping of matter waves through ponderomotive forces and Doppler-mediated photon scattering. Likewise, light interaction with free electrons has recently emerged as a versatile approach to modulate the electron wave function for applications in ultrafast electron microscopy. Here, we combine these two worlds, theoretically demonstrating that matter waves can be optically manipulated via inelastic interactions with optical fields. This allows us to modulate the translational part of the wave function and produce temporally and spatially compressed atomic beam pulses. We realize such modulation through stimulated photon absorption and emission by atoms traversing phase-matching evanescent optical fields generated upon light scattering by a nanostructure and via stimulated Compton scattering in free space without any assistance from material media. Our results support optical manipulation of matter waves as a powerful tool for microscopy, spectroscopy, and exploration of fundamental phenomena associated with light-atom interactions.

Ultrafast Transverse Modulation of Free Electrons by Interaction with Shaped Optical Fields

Spatiotemporal electron-beam shaping is a bold frontier of electron microscopy. Over the past decade, shaping methods evolved from static phase plates to low-speed electrostatic and magnetostatic displays. Recently, a swift change of paradigm utilizing light to control free electrons has emerged. Here, we experimentally demonstrate arbitrary transverse modulation of electron beams without complicated electron-optics elements or material nanostructures, but rather using shaped light beams. On-demand spatial modulation of electron wavepackets is obtained via inelastic interaction with transversely shaped ultrafast light fields controlled by an external spatial light modulator. We illustrate this method for the cases of Hermite-Gaussian and Laguerre-Gaussian modulation and discuss their use in enhancing microscope sensitivity. Our approach dramatically widens the range of patterns that can be imprinted on the electron profile and greatly facilitates tailored electron-beam shaping.

Asynchronous Inelastic Scattering of Electrons at the Ponderomotive Potential of Optical Waves

We study free electron dynamics during inelastic interaction with the ponderomotive potential of a traveling optical wave using classical and quantum-mechanical models. We show that in the strong interaction regime, the electrons trapped in the periodic potential oscillate leading to periodic revolutions of sharp peaks of the density distributions in the real and momentum spaces. In this regime, the synchronicity between the velocity of the optical wave and the electron propagation velocity is not required. Asynchronous interaction enables acceleration or deceleration of a significant fraction of the electrons to a final spectrum with a relative spectral width of 0.5%-2.5%. This technique allows one to accelerate electrons from rest to keV energies while reaching a narrow spectrum of kinetic energies and femtosecond pulsed operation.

Integrated photonics enables continuous-beam electron phase modulation

Integrated photonics facilitates extensive control over fundamental light-matter interactions in manifold quantum systems including atoms1, trapped ions2,3, quantum dots4 and defect centres5. Ultrafast electron microscopy has recently made free-electron beams the subject of laser-based quantum manipulation and characterization6-11, enabling the observation of free-electron quantum walks12-14, attosecond electron pulses10,15-17 and holographic electromagnetic imaging18. Chip-based photonics19,20 promises unique applications in nanoscale quantum control and sensing but remains to be realized in electron microscopy. Here we merge integrated photonics with electron microscopy, demonstrating coherent phase modulation of a continuous electron beam using a silicon nitride microresonator. The high-finesse (Q0 ≈ 106) cavity enhancement and a waveguide designed for phase matching lead to efficient electron-light scattering at extremely low, continuous-wave optical powers. Specifically, we fully deplete the initial electron state at a cavity-coupled power of only 5.35 microwatts and generate >500 electron energy sidebands for several milliwatts. Moreover, we probe unidirectional intracavity fields with microelectronvolt resolution in electron-energy-gain spectroscopy21. The fibre-coupled photonic structures feature single-optical-mode electron-light interaction with full control over the input and output light. This approach establishes a versatile and highly efficient framework for enhanced electron beam control in the context of laser phase plates22, beam modulators and continuous-wave attosecond pulse trains23, resonantly enhanced spectroscopy24-26 and dielectric laser acceleration19,20,27. Our work introduces a universal platform for exploring free-electron quantum optics28-31, with potential future developments in strong coupling, local quantum probing and electron-photon entanglement.

Electron Pulse Compression with Optical Beat Note

Compressing electron pulses is important in many applications of electron beam systems. In this study, we propose to use optical beat notes to compress electron pulses. The beat frequency is chosen to match the initial electron pulse duration, which enables the compression of electron pulses with a wide range of durations. This functionality extends the optical control of electron beams, which is important in compact electron beam systems such as dielectric laser accelerators. We also find that the dominant frequency of the electron charge density changes continuously along its drift trajectory, which may open up new opportunities in coherent interaction between free electrons and quantum or classical systems.

Imprinting the quantum statistics of photons on free electrons

[Figure: see text].

Quantum Wave-Particle Duality in Free-Electron-Bound-Electron Interaction

We present a comprehensive relativistic quantum-mechanical theory for interaction of a free electron with a bound electron in a model, where the free electron is represented as a finite-size quantum electron wave packet (QEW) and the bound electron is modeled by a quantum two-level system (TLS). The analysis reveals the wave-particle duality nature of the QEW, delineating the point-particle-like and wavelike interaction regimes and manifesting the physical reality of the wave function dimensions when interacting with matter. This QEW size dependence may be used for interrogation and coherent control of superposition states in a TLS and for enhancement of cathodoluminescence and electron energy-loss spectroscopy in electron microscopy.

Toward Atomic-Resolution Quantum Measurements with Coherently Shaped Free Electrons

Free electrons provide a powerful tool for probing material properties at atomic resolution. Recent advances in ultrafast electron microscopy enable the manipulation of free-electron wave functions using laser pulses. It would be of great importance if one could combine the spatial resolution of electron microscopes with the ability of laser pulses to probe coherent phenomena in quantum systems. To this end, we propose a novel concept that leverages free electrons that are coherently shaped by laser pulses to measure quantum coherence in materials. We develop the quantum theory of interactions between shaped electrons and arbitrary qubit states in materials, and show how the postinteraction electron energy spectrum enables measuring the qubit state (on the Bloch sphere) and the decoherence or relaxation times (T_{2}/T_{1}). Finally, we describe how such electrons can detect and quantify superradiance from multiple qubits. Our scheme can be implemented in ultrafast transmission electron microscopes (UTEM), opening the way toward the full characterization of the state of quantum systems at atomic resolution.

Modulation of Cathodoluminescence Emission by Interference with External Light

Spontaneous processes triggered in a sample by free electrons, such as cathodoluminescence, are commonly regarded and detected as stochastic events. Here, we supplement this picture by showing through first-principles theory that light and free-electron pulses can interfere when interacting with a nanostructure, giving rise to a modulation in the spectral distribution of the cathodoluminescence light emission that is strongly dependent on the electron wave function. Specifically, for a temporally focused electron, cathodoluminescence can be canceled upon illumination with a spectrally modulated dimmed laser that is phase-locked relative to the electron density profile. We illustrate this idea with realistic simulations under attainable conditions in currently available ultrafast electron microscopes. We further argue that the interference between excitations produced by light and free electrons enables the manipulation of the ultrafast materials response by combining the spectral and temporal selectivity of the light with the atomic resolution of electron beams.

Optical Excitations with Electron Beams: Challenges and Opportunities

Free electron beams such as those employed in electron microscopes have evolved into powerful tools to investigate photonic nanostructures with an unrivaled combination of spatial and spectral precision through the analysis of electron energy losses and cathodoluminescence light emission. In combination with ultrafast optics, the emerging field of ultrafast electron microscopy utilizes synchronized femtosecond electron and light pulses that are aimed at the sampled structures, holding the promise to bring simultaneous sub-Å-sub-fs-sub-meV space-time-energy resolution to the study of material and optical-field dynamics. In addition, these advances enable the manipulation of the wave function of individual free electrons in unprecedented ways, opening sound prospects to probe and control quantum excitations at the nanoscale. Here, we provide an overview of photonics research based on free electrons, supplemented by original theoretical insights and discussion of several stimulating challenges and opportunities. In particular, we show that the excitation probability by a single electron is independent of its wave function, apart from a classical average over the transverse beam density profile, whereas the probability for two or more modulated electrons depends on their relative spatial arrangement, thus reflecting the quantum nature of their interactions. We derive first-principles analytical expressions that embody these results and have general validity for arbitrarily shaped electrons and any type of electron-sample interaction. We conclude with some perspectives on various exciting directions that include disruptive approaches to noninvasive spectroscopy and microscopy, the possibility of sampling the nonlinear optical response at the nanoscale, the manipulation of the density matrices associated with free electrons and optical sample modes, and appealing applications in optical modulation of electron beams, all of which could potentially revolutionize the use of free electrons in photonics.

The coherence of light is fundamentally tied to the quantum coherence of the emitting particle

Coherent emission of light by free charged particles is believed to be successfully captured by classical electromagnetism in all experimental settings. However, recent advances triggered fundamental questions regarding the role of the particle wave function in these processes. Here, we find that even in seemingly classical experimental regimes, light emission is fundamentally tied to the quantum coherence and correlations of the emitting particle. We use quantum electrodynamics to show how the particle's momentum uncertainty determines the optical coherence of the emitted light. We find that the temporal duration of Cherenkov radiation, envisioned for almost a century as a shock wave of light, is limited by underlying entanglement between the particle and light. Our findings enable new capabilities in electron microscopy for measuring quantum correlations of shaped electrons. Last, we propose new Cherenkov detection schemes, whereby measuring spectral photon autocorrelations can unveil the wave function structure of any charged high-energy particle.

Optical coherence transfer mediated by free electrons

We theoretically investigate the quantum-coherence properties of the cathodoluminescence (CL) emission produced by a temporally modulated electron beam. Specifically, we consider the quantum-optical correlations of CL produced by electrons that are previously shaped by a laser field. Our main prediction is the presence of phase correlations between the emitted CL field and the electron-modulating laser, even though the emission intensity and spectral profile are independent of the electron state. In addition, the coherence of the CL field extends to harmonics of the laser frequency. Since electron beams can be focused to below 1 Å, their ability to transfer optical coherence could enable the ultra-precise excitation, manipulation, and spectrally resolved probing of nanoscale quantum systems.

Optical Modulation of Electron Beams in Free Space

We exploit free-space interactions between electron beams and tailored light fields to imprint on-demand phase profiles on the electron wave functions. Through rigorous semiclassical theory involving a quantum description of the electrons, we show that monochromatic optical fields focused in vacuum can be used to correct electron beam aberrations and produce selected focal shapes. Stimulated elastic Compton scattering is exploited to imprint the required electron phase, which is proportional to the integral of the optical field intensity along the electron path and depends on the transverse beam position. The required light intensities are attainable in currently available ultrafast electron microscope setups, thus opening the field of free-space optical manipulation of electron beams.

Control of quantum electrodynamical processes by shaping electron wavepackets

Fundamental quantum electrodynamical (QED) processes, such as spontaneous emission and electron-photon scattering, encompass phenomena that underlie much of modern science and technology. Conventionally, calculations in QED and other field theories treat incoming particles as single-momentum states, omitting the possibility that coherent superposition states, i.e., shaped wavepackets, can alter fundamental scattering processes. Here, we show that free electron waveshaping can be used to design interferences between two or more pathways in a QED process, enabling precise control over the rate of that process. As an example, we show that free electron waveshaping modifies both spatial and spectral characteristics of bremsstrahlung emission, leading for instance to enhancements in directionality and monochromaticity. The ability to tailor general QED processes opens up additional avenues of control in phenomena ranging from optical excitation (e.g., plasmon and phonon emission) in electron microscopy to free electron lasing in the quantum regime.

Shaping quantum photonic states using free electrons

It is a long-standing goal to create light with unique quantum properties such as squeezing and entanglement. We propose the generation of quantum light using free-electron interactions, going beyond their already ubiquitous use in generating classical light. This concept is motivated by developments in electron microscopy, which recently demonstrated quantum free-electron interactions with light in photonic cavities. Such electron microscopes provide platforms for shaping quantum states of light through a judicious choice of the input light and electron states. Specifically, we show how electron energy combs implement photon displacement operations, creating displaced-Fock and displaced-squeezed states. We develop the theory for consecutive electron-cavity interactions with a common cavity and show how to generate any target Fock state. Looking forward, exploiting the degrees of freedom of electrons, light, and their interaction may achieve complete control over the quantum state of the generated light, leading to novel light statistics and correlations.

Single-Cycle Optical Control of Beam Electrons

We report the single-cycle optical control of a freely propagating electron beam with an isolated cycle of midinfrared light. In particular, we produce and characterize a modulated electron current with peak-cycle-specific subfemtosecond structure in time. The direct effects of the carrier-envelope phase, amplitude, and dispersion of the optical waveform on the temporal composition, pulse durations, and chirp of the free-space electron wave function demonstrate the subcycle nature of our control. These results and concept may create novel opportunities in free-electron lasers, laser-driven particle accelerators, ultrafast electron microscopy, and wherever else high-energy electrons are needed with the temporal structure of single-cycle light.

Attosecond metrology in a continuous-beam transmission electron microscope

Electron microscopy can visualize the structure of complex materials with atomic and subatomic resolution, but investigations of reaction dynamics and light-matter interaction call for time resolution as well, ideally on a level below the oscillation period of light. Here, we report the use of the optical cycles of a continuous-wave laser to bunch the electron beam inside a transmission electron microscope into electron pulses that are shorter than half a cycle of light. The pulses arrive at the target at almost the full average brightness of the electron source and in synchrony to the optical cycles, providing attosecond time resolution of spectroscopic features. The necessary modifications are simple and can turn almost any electron microscope into an attosecond instrument that may be useful for visualizing the inner workings of light-matter interaction on the basis of the atoms and the cycles of light.

Ultrafast electron diffraction from nanophotonic waveforms via dynamical Aharonov-Bohm phases

Electron interferometry via phase-contrast microscopy, holography, or picodiffraction can provide a direct visualization of the static electric and magnetic fields inside or around a material at subatomic precision, but understanding the electromagnetic origin of light-matter interaction requires time resolution as well. Here, we demonstrate that pump-probe electron diffraction with all-optically compressed electron pulses can capture dynamic electromagnetic potentials in a nanophotonic material with sub-light-cycle time resolution via centrosymmetry-violating Bragg spot dynamics. The origin of this effect is a sizable quantum mechanical phase shift that the electron de Broglie wave obtains from the oscillating electromagnetic potentials within less than 1 fs. Coherent electron imaging and scattering can therefore reveal the electromagnetic foundations of light-matter interaction on the level of the cycles of light.

High-repetition-rate mid-IR femtosecond pulse synthesis from two mid-IR CW QCL-seeded OPAs

Coherent pulse synthesis in the mid-infrared (mid-IR) domain is of great interest to achieve broadband sources from parent pulses, motivated by the advantages of optical frequency properties for molecular spectroscopy and quantum dynamics. We demonstrate a simple mid-IR coherent synthesizer based on two high-repetition-rate optical parametric amplifiers (OPAs) at nJ-level pump energy. The relative carrier envelope phase between the two OPAs was passively stable for a shared continuous wave (CW) quantum cascade laser (QCL) seed. Lastly, we synthesized mid-IR pulses with a duration of 105 fs ranging from 3.4 to 4.0 µm. The scheme demonstrated the potential to obtain broader mid-IR sources by coherent synthesis from multiple CW QCL-seeded OPAs.

Controlling free electrons with optical whispering-gallery modes

Free-electron beams are versatile probes of microscopic structure and composition^1,2, and have revolutionized atomic-scale imaging in several fields, from solid-state physics to structural biology³. Over the past decade, the manipulation and interaction of electrons with optical fields have enabled considerable progress in imaging methods⁴, near-field electron acceleration^5,6, and four-dimensional microscopy techniques with high temporal and spatial resolution⁷. However, electron beams typically couple only weakly to optical excitations, and emerging applications in electron control and sensing^8-11 require large enhancements using tailored fields and interactions. Here we couple a free-electron beam to a travelling-wave resonant cavity mode. The enhanced interaction with the optical whispering-gallery modes of dielectric microresonators induces a strong phase modulation on co-propagating electrons, which leads to a spectral broadening of 700 electronvolts, corresponding to the absorption and emission of hundreds of photons. By mapping the near-field interaction with ultrashort electron pulses in space and time, we trace the lifetime of the the microresonator following a femtosecond excitation and observe the spectral response of the cavity. The natural matching of free electrons to these quintessential optical modes could enable the application of integrated photonics technology in electron microscopy, with broad implications for attosecond structuring, probing quantum emitters and possible electron-light entanglement.

Coherent interaction between free electrons and a photonic cavity

Advances in the research of interactions between ultrafast free electrons and light have introduced a previously unknown kind of quantum matter, quantum free-electron wavepackets^1-5. So far, studies of the interactions of cavity-confined light with quantum matter have focused on bound electron systems, such as atoms, quantum dots and quantum circuits, which are considerably limited by their fixed energy states, spectral range and selection rules. By contrast, quantum free-electron wavepackets have no such limits, but so far no experiment has shown the influence of a photonic cavity on quantum free-electron wavepackets. Here we develop a platform for multidimensional nanoscale imaging and spectroscopy of free-electron interactions with photonic cavities. We directly measure the cavity-photon lifetime via a coherent free-electron probe and observe an enhancement of more than an order of magnitude in the interaction strength relative to previous experiments of electron-photon interactions. Our free-electron probe resolves the spatiotemporal and energy-momentum information of the interaction. The quantum nature of the electrons is verified by spatially mapping Rabi oscillations of the electron spectrum. The interactions between free electrons and cavity photons could enable low-dose, ultrafast electron microscopy of soft matter or other beam-sensitive materials. Such interactions may also open paths towards using free electrons for quantum information processing and quantum sensing. Future studies could achieve free-electron strong coupling^6,7, photon quantum state synthesis⁸ and quantum nonlinear phenomena such as cavity electro-optomechanics⁹.

Observation of the Relativistic Reversal of the Ponderomotive Potential

The secular dynamics of a nonrelativistic charged particle in an electromagnetic wave can be described by the ponderomotive potential. Although ponderomotive electron-laser interactions at relativistic velocities are important for emerging technologies from laser-based particle accelerators to laser-enhanced electron microscopy, the effects of special relativity on the interaction have only been studied theoretically. Here, we use a transmission electron microscope to measure the position-dependent phase shift imparted to a relativistic electron wave function when it traverses a standing laser wave. The kinetic energy of the electrons is varied between 80 and 300 keV, and the laser standing wave has a continuous-wave intensity of 175  GW/cm^{2}. In contrast to the nonrelativistic case, we demonstrate that the phase shift depends on both the electron velocity and the wave polarization, confirming the predictions of a quasiclassical theory of the interaction. Remarkably, if the electron's speed is greater than 1/sqrt[2] of the speed of light, the phase shift at the electric field nodes of the wave can exceed that at the antinodes. In this case there exists a polarization such that the phase shifts at the nodes and antinodes are equal, and the electron does not experience Kapitza-Dirac diffraction. Our results thus provide new capabilities for coherent electron beam manipulation.

Free-Electron-Bound-Electron Resonant Interaction

Here we present a new paradigm of free-electron-bound-electron resonant interaction. This concept is based on a recent demonstration of the optical frequency modulation of the free-electron quantum electron wave function (QEW) by an ultrafast laser beam. We assert that pulses of such QEWs correlated in their modulation phase, interact resonantly with two-level systems, inducing resonant quantum transitions when the transition energy ΔE=ℏω_{21} matches a harmonic of the modulation frequency ω_{21}=nω_{b}. Employing this scheme for resonant cathodoluminescence and resonant EELS combines the atomic level spatial resolution of electron microscopy with the high spectral resolution of lasers.

Generation and Characterization of Attosecond Microbunched Electron Pulse Trains via Dielectric Laser Acceleration

Dielectric laser acceleration is a versatile scheme to accelerate and control electrons with the help of femtosecond laser pulses in nanophotonic structures. We demonstrate here the generation of a train of electron pulses with individual pulse durations as short as 270±80  attoseconds (FWHM), measured in an indirect fashion, based on two subsequent dielectric laser interaction regions connected by a free-space electron drift section, all on a single photonic chip. In the first interaction region (the modulator), an energy modulation is imprinted on the electron pulse. During free propagation, this energy modulation evolves into a charge density modulation, which we probe in the second interaction region (the analyzer). These results will lead to new ways of probing ultrafast dynamics in matter and are essential for future laser-based particle accelerators on a photonic chip.

All-Optical Scheme for Generation of Isolated Attosecond Electron Pulses

We theoretically propose an all-optical scheme for generation of isolated electron pulses with subfemtosecond durations for ultrafast electron microscopy and diffraction. The scheme is based on simultaneous longitudinal and transverse momentum modulation of freely propagating electrons at two distinct frequencies via ponderomotive interaction with tailored light fields of three phase-controlled laser pulses in vacuum. After a drift distance, an attosecond electron pulse train is formed with the individual pulses displaced in the transverse direction. Spatial filtering of the electron beam by an aperture leads to isolation of an individual attosecond pulse with a duration of <100  as. Subfemtosecond isolated electron pulses will enable direct space-time imaging of attosecond electronic dynamics in atoms, molecules, and solids with atomic spatial resolution, allowing us to directly observe phenomena such as electron tunneling or electron-electron interaction in strong laser fields of few-cycle laser pulses.

Laser phase plate for transmission electron microscopy

Transmission electron microscopy (TEM) of rapidly frozen biological specimens, or cryo-EM, would benefit from the development of a phase plate for in-focus phase contrast imaging. Several types of phase plates have been investigated, but rapid electrostatic charging of all such devices has hindered these efforts. Here, we demonstrate electron phase manipulation with a high-intensity continuous-wave laser beam, and utilize it as a phase plate for TEM. We demonstrate the laser phase plate by imaging an amorphous carbon film. The laser phase plate provides a stable and tunable phase shift without charging or unwanted electron scattering. These results suggest the possibility for dose-efficient imaging of unstained biological macromolecules and cells.

Entanglements of Electrons and Cavity Photons in the Strong-Coupling Regime

This Letter sets a road map towards an experimental realization of strong coupling between free electrons and photons and analytically explores entanglement phenomena that emerge in this regime. The proposed model unifies the strong-coupling predictions with known electron-photon interactions. Additionally, this Letter predicts a non-Columbic entanglement between freely propagating electrons. Since strong coupling can map entanglements between photon pairs onto photon-electron pairs, it may harness electron beams for quantum communication, thus far exclusive to photons.

Anomalous Photon-Induced Near-Field Electron Microscopy

We reveal the classical and quantum regimes of free electron interaction with radiation, common to the general variety of radiation sources (e.g., a Smith-Purcell radiation), the dielectric laser accelerator, and photo-induced near-field electron microscopy (PINEM). Modeling the electron with initial conditions of a coherent quantum electron wave packet, its topology in phase space uniquely defines a universal distinction of three interaction regimes: point-particle-like acceleration, a quantum wave function (PINEM), and a newly reported regime of anomalous PINEM (APINEM). The quantum interference beat of APINEM is capable of improving the spectral resolution of postselective electron microscopy. The particle-wave duality transition between regimes reveals the history-dependent nature of quantum electron interaction with light.

Microscopy and diffraction with attosecond electron pulse trains

Attosecond imaging with electron beams can access optical-field-driven electron dynamics in space and time. Here we report first diffraction and microscopy experiments with attosecond electron pulses. We study attosecond-level timing of Bragg-spot emission and visualize light-wave propagation in space and time.

From strong-field physics in and at nanoscale matter to photonics-based laser accelerators

New ways of controlling quasi-free and free electrons by means of phase-controlled ultrashort laser pulses are demonstrated: from strong-field physics in the conducting 2-d material graphene and at the surface of nanostructures, to laser acceleration of free electrons with a nanophotonic structure, and the demonstration of the longitudinal Kapitza-Dirac effect.

Tilted Electron Pulses

We report the all-optical generation and characterization of tilted electron pulses by means of single-cycle terahertz radiation at an electron-transmitting mirror at slanted orientation. Femtosecond electron pulses with a chosen tilt angle are produced at an almost arbitrary target location. The experiments along with theory further reveal that the pulse front tilt in electron optics is directly connected to angular dispersion. Quantum mechanical considerations suggest that this relation is general for particle beams at any degree of coherence. These results indicate that ultrashort electron pulses can be shaped in space and time as versatilely as femtosecond laser pulses, but at 10^{5} times finer wavelength and subnanometer imaging resolution.