Elastic electron scattering is one of the primary means of investigating materials on the atomic scale. It is usually described in a one-particle approach (the probe electron's evolution in a perturbative static potential), whereas we are dealing here with a two-body between the probe and the sample, both described by separate quantum states, inducing entanglement. In this work, we present a quantum treatment of elastic electron scattering. We find that the entanglement between probe and scatterer can have far-reaching consequences, particularly on coherence and image contrast. As a timely example, we discuss decoherence in Bragg scattering on nanoparticles. We find that conventional scattering theory is recovered in most cases. The situation changes dramatically for freely evolving nanoparticles as, e.g., levitated motional ground states, an active field of research.
Elastic electron scattering is one of the primary means of investigating materials on the atomic scale. It is usually described in a one-particle approach (the probe electron's evolution in a perturbative static potential), whereas we are dealing here with a two-body between the probe and the sample, both described by separate quantum states, inducing entanglement. In this work, we present a quantum treatment of elastic electron scattering. We find that the entanglement between probe and scatterer can have far-reaching consequences, particularly on coherence and image contrast. As a timely example, we discuss decoherence in Bragg scattering on nanoparticles. We find that conventional scattering theory is recovered in most cases. The situation changes dramatically for freely evolving nanoparticles as, e.g., levitated motional ground states, an active field of research.
Matter-wave interferometry is a phenomenon which has been utilized and studied in electron microscopy since the early days of electron optics, with the development of phase contrast imaging, electron diffraction, electron holography, etc. When an electron diffracts off of a crystal lattice, it enters a superposition state of eligible Bragg diffraction components; due to momentum conservation, the crystalline sample must receive an equal and opposite superposition of momentum transfers. In theory, this results in an entangled electron-sample state. Typically, however, the sample is extremely large relative to the electron and is held fast in a sample holder, and this reciprocal momentum transfer is unobservable [1]. We present a proposal [2] to establish a system where such momentum kicks would be resolvable, to enable probing of the matter-wave properties of the sample as well as the free electron. A crystalline silicon nanoparticle is levitated in the sample plane, then released from its trap to enter free-fall. A single electron impacts the nanoparticle and undergoes Bragg diffraction. The nanoparticle receives a momentum transfer from the diffraction and is allowed to continue free-falling for a fixed amount of time before recapture and measurement of its position, X. In synchrony, the position of the electron, x, is measured. Assuming perfect measurement resolution, we calculate the resulting interference pattern for a quantum interaction, where the nanoparticle can behave as a quantum wave (Fig. 1a), or for the case where it behaves as a classical grating aperture (Fig. 1b) for different recapture times, t. The quantum scenario presents additional fringes visible at specific recapture times. Such an experiment provides a test of the matter-wave behaviour of high mass nanoparticles, with an efficient duty cycle and a short interference time, and would allow access to electron-sample entanglement in a transmission electron microscope.
Abstract Quantum entanglement is a basic ingredient in the formalism of quantum mechanics and was held by Heisenberg as its most emblematic feature. But although all its consequences could be foreseen from the start, it took half a century before experiments were carried out to test its more intriguing predictions. One reason was that sufficiently sensitive tools, like lasers, were not yet available, but there was also a lack of interest and imagination among theoreticians, who did not realize its full potential. When experiments were finally realized and opened a ‘weird and ghostlike quantum world’, they were first met with surprise and doubt. In the first experiments on entanglement it was purposely created (using chain-decays or down-conversion), but entanglement is actually a ubiquitous phenomenon appearing in most physical processes, although surviving over so short time-scales that it is usually not observed. The present work deals with ‘measurement-induced entanglement’ in scattering, and starts with a process where its effects can actually be studied in the sub-femtosecond range in a time-resolved manner, the Compton scattering of neutrons. Based on results from this method, the conditions for observing similar effects in other scattering processes are discussed. First, the mechanisms for entanglement creation by a scattering particle are mentioned, then the conditions for observing measurable consequences of it on scattering cross-sections: a very limited number of scattering centers, large zero-point motions of the scatterers and no strong interaction with environment which leads to decoherence.
In order to assess inelastic effects on two fermion entanglement production, we address an exactly solvable two-particle scattering problem where the target is an excitable scatterer. Useful entanglement, as measured by the two particle concurrence, is obtained from post-selection of oppositely scattered particle states. The S matrix formalism is generalized in order to address nonunitary evolution in the propagating channels. We find the striking result that inelasticity can actually increase concurrence as compared to the elastic case by increasing the uncertainty of the single particle subspace. Concurrence zeros are controlled by either single particle resonance energies or total reflection conditions that ascertain precisely one of the electron states. Concurrence minima also occur and are controlled by entangled resonance situations where the electron becomes entangled with the scatterer, and thus does not give up full information of its state. In this model, exciting the scatterer can never fully destroy phase coherence due to an intrinsic limit to the probability of inelastic events.
Atomic resolution transmission electron microscopy (TEM) is an indispensable tool for interdisciplinary investigations of radiation soft or hard materials that evolves rapidly because emerging technologies enable remarkable new experiments [1, 2]. Nowadays, many of them embrace the time domain [2-4]. Beyond materials research, electron microscopy is uniquely suited to study fundamental quantum-mechanical aspects. In fact, it remains concerning that aspects addressing the wave-particle duality, time-dependent measurements, or the entanglement of wave functions and their collapse at a detector still require detailed investigations to guide progress. Certainly, this is no academic matter because such aspects become the bottleneck for the deployment of microscopes that enable studying dynamic processes at high spatiotemporal resolution. It is established practice to describe the scattering of electron ensembles by either a particle picture (Figure 1a) or a wave picture (Figure 1b). Sometimes both pictures are mixed. In spite of calls for solving Schrödinger’s time dependent equation [5], the static Schrödinger equation of wave functions is almost exclusively used to reliably simulate the image formation process using multislice calculations for coherent-elastically scattered electron ensembles [6, 7]. Moreover, the Copenhagen convention of quantum mechanics [8] is adopted, which emphasizes the importance of an observation for the collapse of wave functions (Figure 1b). In order to foster further insight into fundamental quantum-mechanical aspects we pursue a series of experiments with low electron dose rates that guarantee the presence of single electrons in the microscope column at any given time [9]. Consequently, we address the scattering of single electrons (Figure 1c) and build up intensities from electron ensembles over time. In such conditions energy filtered TEM recently revealed the presence of image intensity at sample/vacuum interfaces that cannot be simulated using the established (static) calculation tools even if frozen phonon excitations are considered [10]. In contrast, these intensities can be explained by the time-dependent Schrödinger equation assuming that an energy exchange ΔE occurs during any interaction of the electron with the sample in the Heisenberg limit ΔE = ℏ/2 Δt−1 (coherent-inelastic scattering). In this process a wave package is formed by the superposition of partial waves and a decoherence phase of 0.5-1 radian [10]. The time-dependent Schrödinger equation yields two parameters that characterize the static and the dynamic behavior of any wave function ψ(r, t) = ψ(r) ϕ(t), namely xstatic = ℏ/(2mΔE)1/2 and xdynamic = ℏc/ΔE [11]. The former is the de Broglie wavelength if ΔE is a kinetic energy and the latter becomes an expression for self-coherence if the scattering process is coherent-inelastic in the Heisenberg limit. In these circumstances, the product of Planck’s constant and the speed of light hc is given by the product of the expression for temporal coherence λ2/Δλ and the energy loss ΔE (hc = (λ2/Δλ) ΔE). The validity of the relation is shown in Figure 2 [11]. Thus, the temporal part of Schrödinger’s equation adds contributions of a finite self-coherence length (or time if the electron velocity is considered) to any scattering process, which is neglected if only the static expression is exploited. Further, the formation of wave packages rationalizes the wave-particle duality. The width of wave packages is described by a self-coherence length ls that can vary by over 40 orders of magnitude depending on the strength of interactions [11]. For Coulomb interactions, ls explains the vanishing phase contrast from EF-HRTEM images when the energy loss exceeds values of ΔE > 200 eV [11]. A procedure to experimentally determine the self-coherence length was established earlier by measuring the energy dependent localization of scattered matter waves in surface proximity [10]. Therefore, electron microscopy can be used as a general tool to measure the self-coherence length in real space images on a scale of nanometers and relate this these values to a self-coherence time that reaches attoseconds [12]. Particle and wave descriptions of electron scattering. a) Scattering of electron ensembles in a particle picture. The process is often described by Feynman diagrams, which involves the emission of a virtual photon of lifetime given by Heisenbergs Uncertainty Principle. b) Scattering of electron ensembles in a wave picture that underlies the multi-slice calculations. Infinite wave functions entangle source, sample and detector. The Copenhagen convention requires a wave function collapse upon observation. c) Proposed scattering process of a single electron wave function. The energy exchange occurs in the Heisenberg limit ΔEloss = E2-E1= ℏ/2 (t2-t1)−1= ℏ/2 Δt−1 when a wave package is formed during Δt because of existing energy (wavelength) differences ΔEloss of partial waves [11]. Measurements of self-coherence length and the related self-coherence time at the calculated Heisenberg limit. The colored background marks the quantum realm. The solid lines represent the Heisenberg limit in space and time, which includes the constant ℏc [11].
Interaction of the probe with the specimen in an electron microscope inevitably leads to entanglement between the probe and the scatterer. In spite of the importance of entanglement in many areas of modern physics, this subject has not been touched in the literature. Here, we develop some ideas about entanglement in electron microscopy for a number of scattering mechanisms. The relationship between entropy, density matrices, and coherence is discussed. In addition, we explore the questions “Why is Bragg scattering coherent and energy loss incoherent?” and “When does decoherence play a role?” It seems to be possible to measure decoherence on extremely short timescales of ∼10−8s. This is especially important in view of recent developments in ultrafast electron microscopy.