Toward Atomic-Resolution Quantum Measurements with Coherently Shaped Free Electrons

Cathodoluminescence Nanoscopy: State of the Art and Beyond

Cathodoluminescence (CL) nanoscopy is proven to be a powerful tool to explore nanoscale optical properties, whereby free electron beams achieve a spatial resolution far beyond the diffraction limit of light. With developed methods for the control of electron beams and the collection of light, the dimension of information that CL can access has been expanded to include polarization, momentum, and time, holding promise to provide invaluable insights into the study of materials and optical near-field dynamics. With a focus on the burgeoning field of CL nanoscopy, this perspective outlines the recent advancements and applications of this technique, as illustrated by the salient experimental works. In addition, as an outlook for future research, several appealing directions that may bring about developments and discoveries are highlighted.

Optical polarization analogs in inelastic free-electron scattering

Advances in the ability to manipulate free-electron phase profiles within the electron microscope have spurred development of quantum-mechanical descriptions of electron energy loss (EEL) processes involving transitions between phase-shaped transverse states. Here, we elucidate an underlying connection between two ostensibly distinct optical polarization analogs identified in EEL experiments as manifestations of the same conserved scattering flux. Our work introduces a procedure for probing general tensorial target characteristics including global mode symmetries and local polarization.

Free-electron Ramsey-type interferometry for enhanced amplitude and phase imaging of nearfields

The complex range of interactions between electrons and electromagnetic fields gave rise to countless scientific and technological advances. A prime example is photon-induced nearfield electron microscopy (PINEM), enabling the detection of confined electric fields in illuminated nanostructures with unprecedented spatial resolution. However, PINEM is limited by its dependence on strong fields, making it unsuitable for sensitive samples, and its inability to resolve complex phasor information. Here, we leverage the nonlinear, overconstrained nature of PINEM to present an algorithmic microscopy approach, achieving far superior nearfield imaging capabilities. Our algorithm relies on free-electron Ramsey-type interferometry to produce orders-of-magnitude improvement in sensitivity and ambiguity-immune nearfield phase reconstruction, both of which are optimal when the electron exhibits a fully quantum behavior. Our results demonstrate the potential of combining algorithmic approaches with state-of-the-art modalities in electron microscopy and may lead to various applications from imaging sensitive biological samples to performing full-field tomography of confined light.

Imaging the field inside nanophotonic accelerators

Controlling optical fields on the subwavelength scale is at the core of nanophotonics. Laser-driven nanophotonic particle accelerators promise a compact alternative to conventional radiofrequency-based accelerators. Efficient electron acceleration in nanophotonic devices critically depends on achieving nanometer control of the internal optical nearfield. However, these nearfields have so far been inaccessible due to the complexity of the devices and their geometrical constraints, hampering the design of future nanophotonic accelerators. Here we image the field distribution inside a nanophotonic accelerator, for which we developed a technique for frequency-tunable deep-subwavelength resolution of nearfields based on photon-induced nearfield electron-microscopy. Our experiments, complemented by 3D simulations, unveil surprising deviations in two leading nanophotonic accelerator designs, showing complex field distributions related to intricate 3D features in the device and its fabrication tolerances. We envision an extension of our method for full 3D field tomography, which is key for the future design of highly efficient nanophotonic devices.

Efficient coupling between free electrons and the supermode of a silicon slot waveguide

Laser light can modulate the kinetic energy spectrum of free electrons and induce extremely high acceleration gradients, which are instrumental to electron microscopy and electron acceleration, respectively. We present a design scheme for a silicon photonic slot waveguide which hosts a supermode to interact with free electrons. The efficiency of this interaction relies on the coupling strength per photon along the interaction length. We predict an optimum value of 0.4266, resulting in the maximum energy gain of 28.27 keV for an optical pulse energy of only 0.22 nJ and duration 1 ps. The acceleration gradient is 1.05 GeV/m, which is lower than the maximum imposed by the damage threshold of Si waveguides. Our scheme shows how the coupling efficiency and energy gain can be maximized without maximizing the acceleration gradient. It highlights the potential of silicon photonics technology in hosting electron-photon interactions with direct applications in free-electron acceleration, radiation sources, and quantum information science.

Jaynes-Cummings interaction between low-energy free electrons and cavity photons

The Jaynes-Cummings Hamiltonian is at the core of cavity quantum electrodynamics; however, it relies on bound-electron emitters fundamentally limited by the binding Coulomb potential. In this work, we propose theoretically a new approach to realizing the Jaynes-Cummings model using low-energy free electrons coupled to dielectric microcavities and exemplify several quantum technologies made possible by this approach. Using quantum recoil, a large detuning inhibits the emission of multiple consecutive photons, effectively transforming the free electron into a few-level system coupled to the cavity mode. We show that this approach can be used for generation of single photons, photon pairs, and even a quantum SWAP gate between a photon and a free electron, with unity efficiency and high fidelity. Tunable by their kinetic energy, quantum free electrons are inherently versatile emitters with an engineerable emission wavelength. Hence, they pave the way toward new possibilities for quantum interconnects between photonic platforms at disparate spectral regimes.

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.

Superradiant Electron Energy Loss Spectroscopy

We analyze the interaction between a free electron and an ensemble of identical optical emitters. The mutual coherence and correlations between the emitters can enhance the interaction with each electron and become imprinted on its energy spectrum. We present schemes by which such collective interactions can be realized. As a possible application, we investigate free-electron interactions with superradiant systems, showing how electrons can probe the ultrafast population dynamics of superradiance.

Quantum sensing of strongly coupled light-matter systems using free electrons

Strong coupling in light-matter systems is a central concept in cavity quantum electrodynamics and is essential for many quantum technologies. Especially in the optical range, full control of highly connected multi-qubit systems necessitates quantum coherent probes with nanometric spatial resolution, which are currently inaccessible. Here, we propose the use of free electrons as high-resolution quantum sensors for strongly coupled light-matter systems. Shaping the free-electron wave packet enables the measurement of the quantum state of the entire hybrid systems. We specifically show how quantum interference of the free-electron wave packet gives rise to a quantum-enhanced sensing protocol for the position and dipole orientation of a subnanometer emitter inside a cavity. Our results showcase the great versatility and applicability of quantum interactions between free electrons and strongly coupled cavities, relying on the unique properties of free electrons as strongly interacting flying qubits with miniscule dimensions.

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.

Complete Excitation of Discrete Quantum Systems by Single Free Electrons

We reveal a wealth of nonlinear and recoil effects in the interaction between individual low-energy electrons (≲100  eV) and samples comprising a discrete number of states. Adopting a quantum theoretical description of combined free-electron and two-level systems, we find a maximum achievable excitation probability of 100%, which requires specific conditions relating to the coupling strength and the transition symmetry, as we illustrate through calculations for dipolar and quadrupolar modes. Strong recoil effects are observed when the kinetic energy of the probe lies close to the transition threshold, although the associated probability remains independent of the electron wave function even when fully accounting for nonlinear interactions with arbitrarily complex multilevel samples. Our work reveals the potential of free electrons to control localized excitations and delineates the boundaries of such control.

Quantum-Coherent Light-Electron Interaction in a Scanning Electron Microscope

The last two decades experimentally affirmed the quantum nature of free electron wave packets by the rapid development of transmission electron microscopes into ultrafast, quantum-coherent systems. So far, all experiments were restricted to the bounds of transmission electron microscopes enabling one or two photon-electron interaction sites. We show the quantum coherent coupling between electrons and light in a scanning electron microscope, at unprecedentedly low, subrelativistic energies down to 10.4 keV. These microscopes not only afford the yet-unexplored energies from ∼0.5 to 30 keV providing the optimum electron-light coupling efficiency, but also offer spacious and easily configurable experimental chambers for extended, cascaded optical set ups, potentially boasting thousands of photon-electron interaction sites. Our results make possible experiments in electron wave packet shaping, quantum computing, and spectral imaging with low-energy electrons.

Correlating structure, morphology and properties of metal nanostructures by combining single-particle optical spectroscopy and electron microscopy

The nanoscale morphology of metal nanostructures directly defines their optical, catalytic and electronic properties and even small morphological changes can cause significant property variations. On the one hand, this dependence allows for precisely tuning and exploring properties by shape engineering; next to advanced synthesis protocols, post-synthesis modification through tailored laser modification has become an emerging tool to do so. On the other hand, with this interconnection also comes the quest for detailed structure-property correlation and understanding of laser-induced reshaping processes on the individual nanostructure level beyond ensemble averages. With the development of single-particle (ultrafast) optical spectroscopy techniques and advanced electron microscopy such understanding can in principle be gained at the femtosecond temporal and atomic spatial scale, respectively. However, accessing both on the same individual nanostructure is far from straightforward as it requires the combination of optical spectroscopy and electron microscopy. In this Minireview, we highlight key studies from recent years that performed such correlative measurements on the same individual metal nanostructure either in a consecutive ex situ manner or in situ inside the electron microscope. We demonstrate that such a detailed correlation is critical for revealing the full picture of the structure-property relationship and the physics behind light-induced nanostructure modifications. We put emphasis on the advantages and disadvantages of each methodology as well as on the unique information that one can gain only by correlative studies performed on the same individual nanostructure and end with an outlook on possible further development of this field in the near future.

Unitary two-state quantum operators realized by quadrupole fields in the electron microscope

In this work, a novel method for using a set of electromagnetic quadrupole fields is presented to implement arbitrary unitary operators on a two-state quantum system of electrons. In addition to analytical derivations of the required quadrupole and beam settings which allow an easy direct implementation, numerical simulations of realistic scenarios show the feasibility of the proposed setup. This is expected to pave the way not only for new measurement schemes in electron microscopy and related fields but even one day for the implementation of quantum computing in the electron microscope.

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.

Superradiance and Subradiance due to Quantum Interference of Entangled Free Electrons

When multiple quantum emitters radiate, their emission rate may be enhanced or suppressed due to collective interference in a process known as super- or subradiance. Such processes are well known to occur also in light emission from free electrons, known as coherent cathodoluminescence. Unlike atomic systems, free electrons have an unbounded energy spectrum, and, thus, all their emission mechanisms rely on electron recoil, in addition to the classical properties of the dielectric medium. To date, all experimental and theoretical studies of super- and subradiance from free electrons assumed only classical correlations between particles. However, dependence on quantum correlations, such as entanglement between free electrons, has not been studied. Recent advances in coherent shaping of free-electron wave functions motivate the investigation of such quantum regimes of super- and subradiance. In this Letter, we show how a pair of coincident path-entangled electrons can demonstrate either super- or subradiant light emission, depending on the two-particle wave function. By choosing different free-electron Bell states, the spectrum and emission pattern of the light can be reshaped, in a manner that cannot be accounted for by a classical mixed state. We show these results for light emission in any optical medium and discuss their generalization to many-body quantum states. Our findings suggest that light emission can be sensitive to the explicit quantum state of the emitting matter wave and possibly serve as a nondestructive measurement scheme for measuring the quantum state of many-body systems.

Quantum Entanglement and Modulation Enhancement of Free-Electron-Bound-Electron Interaction

The modulation and engineering of the free-electron wave function bring new ingredients to the electron-matter interaction. We consider the dynamics of a free-electron passing by a two-level system fully quantum mechanically and study the enhancement of interaction from the modulation of the free-electron wave function. In the presence of resonant modulation of the free-electron wave function, we show that the electron energy loss and gain spectrum is greatly enhanced for a coherent initial state of the two-level system. Thus, a modulated electron can function as a probe of the atomic coherence. We further find that distantly separated two-level atoms can be entangled through interacting with the same free electron. Effects of modulation-induced enhancement can also be observed using a dilute beam of modulated electrons.