Resilient quantum electron microscopy

We investigate the fundamental limit of biological quantum electron microscopy, which is designed to go beyond the shot noise limit. Inelastic scattering is expected to be the main obstacle in this setting, especially for thick specimens of actual biological interest. Here we describe a measurement procedure that, in principle, significantly neutralizes the effect of inelastic scattering.

Figure 1

Illustrations of quantum states at different stages of spatial-frequency (SF) selective measurement. The electron state, which may have been transferred to a quantum computer, in the branch of the entire wave function, wherein Q1 state is $|\overline{s}\rangle$, is shown in the left figure. In a similar way, one with $|\overline{a}\rangle$ is on the right. See the main text for more information. (a) The state right after transmission of an electron through the specimen. The shape of the “molecule,” which is imprinted on the phase of electron wave function, is shown in dashed curves because it is not “visible,” in a loose analogy with transmission electron microscopy (TEM) imaging. The electron wavefront is such that the wave goes left if Q1 state is $|\overline{s}\rangle$, and goes right if Q1 state is $|\overline{a}\rangle$, as indicated by the arrows. (It reflects our convention of discrete Fourier transform.) (b) We apply QFT to the quantum states $|n,m\rangle$, representing the transmitted array of $M\times M$ beams, to obtain essentially a far-field state. The left-moving wave in (a) goes to point S, while the right-moving one goes to A. Elastically scattered waves, whose intensity is indicated by dashed curves, surround the transmitted waves at S and A. Unlike actual diffraction in real electron optics, the periodic boundary condition (PBC) is applicable in a quantum Fourier transform (QFT). Thus, the wave that goes beyond the left boundary, for example, comes out from the right side. The scattered amplitudes at B and C are the negative of complex conjugate (NCC) to each other. The same applies to D and E due to PBC. The amplitudes of interest at F and G are NCC to itself and hence is pure imaginary. These amplitudes at F and G are “added” to, although the wave function branch is different, the transmitted waves, respectively, at S and A with a $\frac{\pi }{2}$ phase difference. (c) The state after split inverse Fourier transformation (FT), where inverse FT is performed in $n<0$ and $n\ge 0$ regions separately. If the outcome of Q2 measurement in Step 7 is $c=0$, the $n<0$ part of the map is left available, while for $c=1$, the $n\ge 0$ part remains. In each region, the shape of the “molecule” is recovered to an extent. In contrast to conventional in-focus phase contrast microscopy, we convert $un$ wanted phase into amplitude in Step 6 before the split inverse FT, because we want to accumulate the signal phase onto Q1. (In Step 12, we measure Q1 with respect to basis states $\left\{\right|\overline{\uparrow }\rangle ,|\overline{\downarrow }\rangle \}$, thus finally “converting the phase into amplitude,” figuratively speaking, when the final measurement is done.) Since the unwanted information is converted to amplitude and is “visible.” again in a loose analogy with TEM, the “molecules” are drawn with solid curves, as opposed to dashed curves that are found in (a). The images of the “molecule” is shown dimly when it is high-pass filtered, where the signal is indeed expected to be weak.

Figure 2

Inelastic scattering neutralization (ISN). (a) A state appearing in inelastic scattering neutralization. We schematically show the inelastically scattered wave function ${\mathrm{\Psi }}_0\left(\mathbf{k}\right)$ in the far field, which is an odd function, with unknown dipole orientation $\mathbf{a}$ (here $\mathbf{a}$ points towards the upper-right direction). The white and gray regions indicate positive and negative regions of ${\mathrm{\Psi }}_0\left(\mathbf{k}\right)$, although such a characterization is oversimplified because the wavefront should also have the phase factor $e^{-i\mathbf{q}·{\mathbf{r}}_0}$ that represents tilt. As is the case with elastic scattering, the amplitudes at B and C, or D and E, are the negative of complex conjugate (NCC) to each other. This helps to generate $\eta \in \mathbb{R}$. (b) An averaged intensity map of inelastically scattered electron waves in the far field. This is one branch of the far field wherein Q1 state is $|\overline{s}\rangle$. Although the far field has a finite area (call it a cell) with the periodic boundary condition (PBC), here it is shown in a expanded way. The intensity of inelastically scattered electron wave, which is averaged over all dipole orientations $\mathbf{a}$, is shown in dashed circles. Although this intensity pattern is repeated due to PBC, only the one originating from the point S of a single cell is shown for clarity. The white regions correspond to the set $\mathcal{S}$ described in the text, while the gray regions correspond to $\mathcal{A}$. If one focuses on an area of either $\mathcal{S}$ or $\mathcal{A}$ type in a particular cell, then one should see that the wave intensity in that area comes from waves originating from many different cells. Subsequently, waves in ${\mathcal{M}}_s$ and ${\mathcal{M}}_a$ regions, respectively, are mixed up within these regions in the cell during the split inverse FT. Equations (122) and (124) express incoherent addition of these waves. (c) This plot shows how $\mu \approx \left|\eta \right|$, which quantifies nonideality of ISN, changes with $\beta =\frac{\lambda }{2\sigma }$, where $\sigma$ is the resolution of interest. Things get better at higher resolution.

Figure 3

The optimal repetition number $k$, enabled by inelastic scattering neutralization, is plotted against $\beta =\lambda /2\sigma$, where $\sigma$ is resolution of interest. For example, $\beta =2\mathrm{mrad}$ corresponds to $\sigma =0.5\mathrm{nm}$. The phase resolution is improved by $\approx 0.65\sqrt{k}$. The dash-dotted curve and the solid curve correspond to $k_{\mathrm{opt}}$ defined in the main text for $\mathrm{\Lambda }/t=10$ and 5, which roughly correspond to the specimen thickness of $t\approx 30\mathrm{nm}$ and $60\mathrm{nm}$, respectively. On the other hand, radiation damage governs the allowed number of electrons $N_{\mathrm{sq}}$ (the dotted curve) and the actual repetition number $k$ should equal $min\{k_{\mathrm{opt}},N_{\mathrm{sq}}\}$.

Figure 4

Simulated images of the Marburg virus VP35 domain molecule [24]. (a) A bandpass-filtered phase map. (b) Imaging without quantum enhancement, i.e., the repetition number $k=1$. (c) QEM imaging without inelastic scattering neutralization (ISN), wherein $k_1=\mathrm{\Lambda }/t=10$, roughly corresponding to a specimen thickness $t\approx 30\mathrm{nm}$. (d) QEM imaging with ISN. The repetition number is ${\tilde{k}}_2$, where $\mathrm{\Lambda }/t=10$. (e) QEM imaging without ISN, wherein $\mathrm{\Lambda }/t=5$ (i.e., specimen thickness $t\approx 60\mathrm{nm}$). (f) QEM imaging with ISN, wherein $\mathrm{\Lambda }/t=5$. The horizontal length of all images is $10\mathrm{nm}$.