Measurement of Self-coherence in the Transmission Electron Microscope for Detecting Time-dependent Coherent-inelastic Electron Scattering in the Heisenberg Limit
Christian F. Kisielowski, Peta Specht, Joerg Ralf Jinschek +1 more
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].
Microscopy and Microanalysis•Jul 1, 2025•0 citations