Bayesian Optimization of Interaction Free Measurement with Electrons

Abstract

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.