Abstract Reducing radiation damage is becoming a crucial factor in the development of future electron microscopes. The concept of interaction-free imaging utilizing a Mach–Zehnder interferometer has been proposed originally in. 1993 by Elitzur and Vaidman. Using the quantum Zeno effect the quantum efficiency for interaction-free measurements can be increased to near 100 % with increasing number of interrogations provided the sample is a perfect (thick) absorber. However, for an electron beam the sample is in general semi-transparent depending on the thickness. In electron microscopy the interaction-free experiment is complicated by the presence of elastic and inelastic scattering processes in the electron-sample interaction. The ratio of inelastically versus elastically scattered electrons is determined by the nature of the sample and the thickness and cannot be reduced arbitrarily. Because inelastically scattered electrons transfer energy to the sample and cause radiation damage this puts a limit to the elastically scattered electrons and consequently to the resolution of electron imaging. In this paper, the inelastic scattering is modeled as an “absorption” function on the modulus of the electron wave. Image simulations and Argand plots are used to investigate the convergence rate of the Zeno iterations versus different type of samples, in particular a double strained DNA helical sample. It is found that the stronger objects reach maximum quantum efficiency faster than weaker objects for which more interrogations are needed to reach the maximum quantum efficiency. For a general hybrid amplitude/phase object the “phase” information tends to converge to zero as the number of interrogation increases.
Abstract Reducing radiation damage is becoming a crucial factor in the development of future electron microscopes. The concept of interaction-free imaging utilizing a Mach–Zehnder interferometer has been proposed originally in. 1993 by Elitzur and Vaidman. Using the quantum Zeno effect the quantum efficiency for interaction-free measurements can be increased to near 100 % with increasing number of interrogations provided the sample is a perfect (thick) absorber. However, for an electron beam the sample is in general semi-transparent depending on the thickness. In electron microscopy the interaction-free experiment is complicated by the presence of elastic and inelastic scattering processes in the electron-sample interaction. The ratio of inelastically versus elastically scattered electrons is determined by the nature of the sample and the thickness and cannot be reduced arbitrarily. Because inelastically scattered electrons transfer energy to the sample and cause radiation damage this puts a limit to the elastically scattered electrons and consequently to the resolution of electron imaging. In this paper, the inelastic scattering is modeled as an “absorption” function on the modulus of the electron wave. Image simulations and Argand plots are used to investigate the convergence rate of the Zeno iterations versus different type of samples, in particular a double strained DNA helical sample. It is found that the stronger objects reach maximum quantum efficiency faster than weaker objects for which more interrogations are needed to reach the maximum quantum efficiency. For a general hybrid amplitude/phase object the “phase” information tends to converge to zero as the number of interrogation increases.
Interaction-free measurement (IFM) has been proposed as a means of high-resolution, low-damage imaging of radiation-sensitive samples, such as biomolecules and proteins. The basic setup for IFM is a Mach-Zehnder interferometer, and recent progress in nanofabricated electron-diffraction gratings has made it possible to incorporate a Mach-Zehnder interferometer in a transmission electron microscope (TEM). Therefore, the limits of performance of IFM with such an interferometer and a shot-noise limited electron source (such as that in a TEM) are of interest. In this paper, we compare the error probability and sample damage for ideal IFM and classical imaging schemes, through theoretical analysis and numerical simulation. We consider a sample that is either completely transparent or completely opaque at each pixel. In our analysis, we also evaluate the impact of an additional detector for scattered electrons. Inclusion of the scattering detector results in reduction of error by up to an order of magnitude, for both IFM and classical schemes. We also investigate a sample reillumination scheme based on updating priors after each round of illumination and find that this scheme further reduces error. Implementation of these methods is likely achievable with existing instrumentation and would result in improved resolution in low-dose electron microscopy.
High energy electrons enable atomic resolution, but also engender radiolysis or ionization damage to beam-sensitive materials, hampering the atomic imaging of medically and technologically relevant materials/processes such as cellular dynamics and proteomics. The dose required to resolve beam-sensitive materials usually far exceeds its critical dose, requiring dose dispersion across identically prepared samples that incur extensive computational processing and precise sample preparation. Numerous techniques are proposed and employed to minimize ionization damage while maximizing information acquired. The Elitzur and Vaidman quantum bomb detector is one such method [1], involving interaction-free measurements (IFM) [2-3]. In the thought experiment of Elitzur and Vaidman, single particles that are spacelike separated pass through an aligned Mach–Zehnder interferometer such that the particles self-interfere resulting in detections only at the bright port (Fig. 1(a)). Inserting an opaque object into the interferometer, disrupts the self-interference of the particles such that detections occur at the dark port (Fig. 1(b)). These dark port detections (IFMs) inform on the presence of the opaque object without the particles directly interacting with it. This phenomena extends to phase objects enabling quantitative phase imaging using interaction free measurements. A two-grating Mach–Zehnder interferometer (2GMZI) was constructed in a Titan TEM to realize IFM with electrons [4]. Now we construct the 2GMZI in a Spectra TEM similarly as before and is schematically shown in Figure 1(c). There are numerous new features from the previous generation Titan TEM pertinent to 2GMZI—such as a cold FEG and a Dectris ELA (direct electron) detector—improving electron coherence and signal-to-noise of the dark port. In the interim between the 2GMZI setups, analyses and simulations were performed to optimize the two nanofabricated gratings in mill pattern and configuration. Here we discuss such grating optimizations with an objective to maximize the Fisher information per unit radiation damage in beam-sensitive materials. The Fisher information (FIM) of interest quantifies the expected information in the detected electrons about phase differences imparted by the specimen (see Figure 2). The FIM is tunable by varying the proportion of electrons propagating via each path determined by the gratings. For instance, the pump probe configuration—where the pump carries most electrons through vacuum and the probe passes through the sample—minimizes radiation damage to the sample, but reduces the Fisher information and increases background noise via longer imaging sessions to acquire the necessary signal. The output grating may also be tuned to create multiple dark ports, increasing the overall signal. The interplay between minimizing radiation damage while maximizing the Fisher information leads to many local solutions where the global solution status depends on the critical dose of samples and the background shot noise. Utilizing these global solution configurations of 2GMZI for IFM imaging will improve low-dose phase imaging of beam-sensitive materials, decreasing the dose for high resolution imaging [5]. Sub figures (a) and (b) illustrate an electron, Mach–Zehnder interferometer in which traditional optical components—such as beam splitters and mirrors—are interchanged for gratings and magnetic lenses. Inserting an opaque object (b), prevents deconstructive interference at the dark port (DP), enabling interaction free measurements. Sub figure (c) further details the implementation of a Mach–Zehnder interferometer in an electron microscope. These two sub figures were simulated using two incident probes in the sample plane like Figure 1(c), and the specimen-induced phase-difference is swept. Sub figure (a) shows the normalized intensity of the modes, in which mode -1 and 0 correspond to the dark and bright port, respectively. Sub figure (b) is the corresponding Fisher information for each respective mode. The outer modes should be ideally suppressed.
One of the astounding consequences of quantum mechanics is that it allows the detection of a target using an incident probe, with only a low probability of interaction of the probe and the target. This 'quantum weirdness' could be applied in the field of electron microscopy to generate images of beam-sensitive specimens with substantially reduced damage to the specimen. A reduction of beam-induced damage to specimens is especially of great importance if it can enable imaging of biological specimens with atomic resolution. Following a recent suggestion that interaction-free measurements are possible with electrons, we now analyze the difficulties of actually building an atomic resolution interaction-free electron microscope, or "quantum electron microscope". A quantum electron microscope would require a number of unique components not found in conventional transmission electron microscopes. These components include a coherent electron beam-splitter or two-state-coupler, and a resonator structure to allow each electron to interrogate the specimen multiple times, thus supporting high success probabilities for interaction-free detection of the specimen. Different system designs are presented here, which are based on four different choices of two-state-couplers: a thin crystal, a grating mirror, a standing light wave and an electro-dynamical pseudopotential. Challenges for the detailed electron optical design are identified as future directions for development. While it is concluded that it should be possible to build an atomic resolution quantum electron microscope, we have also identified a number of hurdles to the development of such a microscope and further theoretical investigations that will be required to enable a complete interpretation of the images produced by such a microscope.
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].
The major resolution-limiting factor in cryoelectron microscopy of unstained biological specimens is radiation damage by the very electrons that are used to probe the specimen structure. To address this problem, an electron microscopy scheme that employs quantum entanglement to enable phase measurement precision beyond the standard quantum limit has recently been proposed [H. Okamoto, Phys. Rev. A 85, 043810 (2012)]. Here we identify and examine in detail measurement errors that will arise in the scheme. Emphasis is given to considerations concerning inelastic scattering events because, in general, schemes assisted with quantum entanglement are known to be highly vulnerable to lossy processes. We find that the amount of error due to both elastic and inelastic scattering processes is acceptable provided that the electron beam geometry is properly designed and the specimen is sufficiently thin.
We discuss the effect of semitransparency in a quantum-Zeno-like interaction-free measurement setup, a quantum-physics based approach that might significantly reduce sample damage in imaging and microscopy. With an emphasis on applications in electron microscopy, we simulate the behavior of probe particles in an interaction-free measurement setup with semitransparent samples, and we show that the transparency of a sample can be measured in such a setup. However, such a measurement is not possible without losing (i.e., absorbing or scattering) probe particles in general, which causes sample damage. We show how the amount of lost particles can be minimized by adjusting the number of round trips through the setup, and we explicitly calculate the amount of lost particles in measurements which either aim at distinguishing two transparencies or at measuring an unknown transparency precisely. We also discuss the effect of the sample causing phase shifts in interaction-free measurements. Comparing the resulting loss of probe particles with a classical measurement of transparency, we find that interaction-free measurements only provide a benefit in two cases: first, if two semitransparent samples with a high contrast are to be distinguished, interaction-free measurements lose less particles than classical measurements by a factor that increases with the contrast. This implies that interaction-free measurements with zero loss are possible if one of the samples is perfectly transparent. A second case where interaction-free measurements outperform classical measurements is if three conditions are met: the particle source exhibits Poissonian number statistics, the number of lost particles cannot be measured, and the transparency is larger than approximately 21. In all other cases, interaction-free measurements lose as many probe particles as classical measurements or more. Aside from imaging of gray levels, another possible application for interaction-free measurements is the detection of arbitrarily small phase shifts in transparent samples.