QEM: Microwave guides for quantum electron microscopy

Motivation

Electron microscopy has revolutionized our understanding of the very small by its capacity to image with atomic scale resolution, however, its usage for biological specimens is limited by the damage it causes through radiolysis, heat and knock-on damage. [1] Thus, biological specimens only survive a certain electron dose before degrading which sets the attainable resolution for single objects at 5-10 nm [2] since image information is shot-noise limited. Sub-nanometer resolution can only be achieved through the acquisition of thousands of images of identical structures and using a sophisticated and computational expensive reconstruction algorithm in order to generate the 3-D structure. Atomically resolved images of individual, isolated biological specimens with negligible damage to sample is still impossible with the current state-of-the-art electron microscopy. However, implementing measurement techniques based on the quantum mechanical phenomenon known as “interaction-free measurement” could potentially overcome this limitation by reducing the damage and lead to the development of a new instrument: The quantum electron microscope (QEM) [3,4].

Interaction-free-measurement

Interaction-Free measurement (IFM) is based on the idea that an opaque object may be observed by detecting a particle (e.g. photon or electron) that did not interact with that object in a classical sense [5,6]. This concept can be described using a Mach-Zehnder interferometer, shown in the figure below: A fully opaque object one arm of the interferometer destroys interference at the combining element leading to a possible detection of a particle at the output of the interferometer in detector 2.

In this case, one knows that the particle traveled along the lower beam path resulting in an event at detector 2. Since detector 2 never detects an electron, when there is no object in the upper beam path, the presence of the object was detected, although there was no (classical) interaction between the object and the detected particle. Nevertheless, there is still a high probability for the particle to be lost at the opaque object (which could cause damage to it) in this simple Mach-Zehnder interferometer setup. However, this technique was conceptually extended to success probabilities arbitrarily close to one using an approach analogous to a discrete form of the quantum Zeno effect [7].

An interaction-free measurement based on a Mach–Zehnder interferometer.
(a) An incident particle (photon/electron) enters the interferometer from the left and is split into a superposition of two wave-packets by a 50/50 beam-splitter . The two wave-packets then travel through the two arms of the interferometer via mirrors, before entering a second 50/50 beam-splitter. The interferometer geometry is arranged such that a particle is never detected at detector 2 due to deconstructive interference, while a particle is always detected at detector 1.
(b) When an object is placed in the path of the upper arm of the interferometer, deconstructive interference of the two-wave-packets is prevented, and a particle may now be detected at detector 2. In that case, the presence of the object has been determined by detection of a particle that did not (classically) interact with the object.

IFM with high success probability

The approach is based on increasing the reflectivity of the beam-splitters, thus reducing the intensity of the beam directed at the specimen of interest, while extending the system to interrogate the specimen multiple times within a single measurement. A cavity could be used to enable the coherent evolution of intensity transfer between the sample and reference beams to achieve higher detection probabilities while simultaneously reducing the intensity at the specimen. These higher detection probabilities have been experimentally demonstrated for photons [7-9] and subsequently applied to imaging systems [10]. These advanced IFM techniques yield a success probability of one, meaning that a single detected photon will be sufficient to determine the presence of an object in the path of the beam, and that no photon will be absorbed by the object. A partially absorbing object would result in a lower success probability [11] and consequently reduce the benefit of the IFM approach. IFM has yet to be shown for electrons and be implemented in an electron microscope where damage (which IFM can potentially reduce) is the main constraint.

Example of an interaction-free measurement scheme with a success probability of ∼0.61, using an ultra-sensitive bomb as a sample object (if the photon/electron hits the bomb, the bomb explodes). Two cavities are coupled via a beam splitter with reflectivity ρ ≈ 0.905. After N = 5 round trips, a photon/ electron starting in the left cavity is fully transferred to the right cavity, if there is no object. The transfer is nonlinear because the amplitudes in the two cavities add up coherently at the beam splitter. If the second cavity is blocked by the bomb, this coherent build-up can no longer occur and the transfer of amplitude is slowed. An interaction-free detection of the bomb is carried out by measuring whether the photon/electron is in the left or in the right cavity after N = 5 round trips. There is a ∼0.61 probability of detecting the bomb without making it explode.

QEM collaboration funded by Moore foundation

After it was realized that interaction-free measurements could be performed with electrons as well [3], an international collaboration (Link: http://www.rle.mit.edu/qem/) consisting of research groups based in Erlangen, Delft, Boston and Stanford was founded with the help of the Gordon and Betty Moore Foundation (Link: https://www.moore.org/). This QEM collaboration analyzes the difficulties of actually building an atomic resolution interaction-free electron microscope, or quantum electron microscope. This quantum electron microscope would require a number of unique components not found in conventional electron microscopes, which include a coherent electron beam-splitter or two-state-coupler, and a resonator structure to allow each electron to interrogate the specimen multiple times. Different system designs have been proposed [4], we investigate the potential of a two-state-coupler based on the electro-dynamical pseudopotential created by microwave fields by means of properly shaped electrodes on planar substrates and the usage of on-chip mirrors as a resonating structure.

Microwave guides for electrons

The basic principle of the microwave electron guides is based on a linear Paul trap. Paul traps allow trapping of charged particles in a pure, alternating electrical field. Field oscillations that are fast compared to the movement of the particle to be trapped result in a harmonic restoring force averaged over time. The restoring force always accelerates the particle to the minimum of the field and thereby enables stable confinement. Since the charge to mass ration for electrons is very high compared to ions Paul traps were until recently limited to ions. Trapping electrons therefore require high frequencies for changing the polarity of the electrical field. Typical values for these frequencies are in the microwave regime between 1 GHz and 10 GHz. Electrons with an energy of 1 to 10 eV are injected at one end of the conductor and deflected in an angle of 30° by the curved electrodes (as shown in the figure below). For this reason, the guided electrons can be spatially separated from the unguided ones on the detector. Using this, we could show that Paul traps also work for electrons and that guiding of electrons in a two-dimensional confinement is possible [12,13]. Careful electrode design and adjustment of the microwave parameters allows for precise control over the fields and thus, over the motion of the guided electrons.

Going to a different electrode geometry, we were able to show the realization of a new electro-optical instrument: The microwave beam splitter [14].

Electrons follow the curved microwave guide, when the microwave potential is switched on. [9]

Microwave Electron Beam Splitter

In our recent publication [15], we demonstrate that we can divide an electron beam with much higher energy than previously possible into two beams by using a double-layer microwave structure. The working principle of this electron beam splitter is based on the fact that the movement of charged particles in a pure alternating electric field can also be described as the movement in an effective potential for comparatively high frequencies. The electron beam is divided by smoothly dividing the transverse pseudopotential minimum, into which the electrons are injected, along the structure into two minima (see figure below). The electrons follow the evolution of the pseudopotential from a single-well into a double-well and are divided into two electron beams. The single-layer predecessor structures worked for electron energies of up to 4 eV [14], which was extended to 200 eV by the use of two microwave substrates aligned to each other. The improvement by almost two orders of magnitude can be explained by the different geometry, because the double-layer structures have much larger geometric factors for the calculation of the trap frequency and the pseudo-potential depth than the corresponding single-layer structures, which enables the guiding and splitting of a much higher-energy electron beam.

Electrode layout of the two-sided microwave beam splitter for electrons and simulation results of the pseudopotential. The isopotential surface of the pseudopotential is plotted as a transparent blue region. The electron beam is injected into the center in between the chips into the y-direction and is split transversely into two beams as the central microwave electrode widens. [12]

Outlook: Microwave Electron Interferometer and Implementation in an SEM

Future experiments should show adiabatic splitting by matching a diffraction-limited Gaussian electron beam to the transverse ground state of the guiding potential. In the adiabatic limit, an electron wave packet propagates along the guiding potential and remains in the quantum ground state along the guide. Overlapping the two resulting guided and separated electron beams at a distant detector would result in interference stripes. Such a microwave chip-based electron interferometer would herald new quantum optics experiments and may become the amplitude beam splitter needed for the non-invasive quantum electron microscopy concept.

We plan to integrate such a coherent beam splitter into a scanning electron microscope (SEM) in order to realize an quantum electron microscope (as depicted in the figure below): The microwave beam splitter is combined with an on-chip mirror in order to transfer the electron between the reference and sample beam.

An electron is injected into the reference beam. The “barn door” (an electrostatic mirror) traps the electron for a certain number of roundtrips during which part of the amplitude of the electron’s wavefunction is transferred by the microwave beam splitter into the sample beam depending on the opaqueness of the sample at the point the sample beam is focused on.

[1] D.B. Williams and C.B. Carter Transmission Electron Microscopy: A Textbook for Materials Science (2nd ed.), Springer, New York (2009)

[2] C.A. Diebolder, A.J. Koster and R.I. Koning, J. Microsc., 248 (2012), pp. 1-5

[3] Putnam, W. P., Yanik, M. F. Noninvasive electron microscopy with interaction-free quantum measurements. Phys. Rev. A 80, 040902(R) (2009).

[4] Kruit, Pieter, et al. “Designs for a quantum electron microscope.” Ultramicroscopy 164 (2016): 31-45.

[5] Elitzur, A. C., Vaidman, L. Quantum mechanical interaction-free measurements. Foundations of Physics 23, 987–997 (1993)

[6] L. Vaidman, Found. Phys., 33 (2003), pp. 491-510

[7] Kwiat, P., Weinfurter, H., Herzog, T., Zeilinger, A., Kasevich, M. A. Interaction-free measurement. Physical Review Letters 74, 4763–4766 (1995)

[8] T. Tsegaye, E. Goobar, A. Karlsson, G. Björk, M. Loh and K. Lim, Phys. Rev. A, 57 (1998), pp. 3987-3990

[9] J.-S. Jang, Phys. Rev. A, 59 (1999), pp. 2322-2329

[10] White, Andrew G., et al. ““Interaction-free” imaging.” Physical Review A 58.1 (1998): 605.

[11] S. Thomas, C. Kohstall, P. Kruit and P. Hommelhoff, Phys. Rev. A, 90 (2014), p. 053840

[12] Hoffrogge, Johannes; Fröhlich, Roman; Kasevich, Mark A.; Hommelhoff, Peter; Microwave guiding of electrons on a chip. Phys. Rev. Lett. 106, 193001 (2011).

[13] Hoffrogge, Johannes; Hommelhoff, Peter; Planar microwave structures for electron guiding. New J. Phys. 13, 095012 (2011).

[14] J. Hammer, S. Thomas, P. Weber, and P. Hommelhoff, “Microwave chipbased beam splitter for low-energy guided electrons,” Physical Review Letters 114, 254801 (2015).

[15] Zimmermann, R., et al. “Beam splitting of low-energy guided electrons with a two-sided microwave chip.” Applied Physics Letters 115.10 (2019): 104103.