Inelastic Mach-Zehnder Interferometry with Free Electrons

We use a novel scanning electron Mach-Zehnder interferometer constructed in a conventional transmission electron microscope to perform inelastic interferometric imaging with free electrons. An electron wave function is prepared in two paths that pass on opposite sides of a gold nanoparticle, where plasmons are excited before the paths are recombined to produce electron interference. We show that the measured spectra are consistent with theoretical predictions, specifically that the interference signal formed by inelastically scattered electrons is $\pi$ out of phase with respect to that formed by elastically scattered electrons. This technique is sensitive to the phase of localized optical modes, because the interference signal amounts to a substantial fraction of the transmitted electrons. Thus, we argue that inelastic interferometric imaging with our scanning electron Mach-Zehnder interferometer provides a new platform for controlling the transverse momentum of free electrons and studying coherent electron-matter interactions at the nanoscale.

Figure 1

Illustration of experiment. An electron is split by a grating ($G1$) with pitch $\mathrm{\Lambda }=2\pi /|{\mathbf{k}}_0|$, preparing it in the superposition state $|{\psi }_i⟩$. The separated paths are focused and steered by magnetic lenses to interact with the dipolar plasmon of a gold nanoparticle (interaction Hamiltonian $H_I$) producing the final state $|{\psi }_f⟩$. The paths are recombined using a second grating ($G2$ with the same pitch $\mathrm{\Lambda }$) that is continuously translatable by ${\mathbf{x}}_0$. The position ${\mathbf{x}}_0$ of $G2$ tunes the relative path phase in the interferometer output as $\mathrm{\Delta }{\varphi }_{\mathrm{int}}=2{\mathbf{k}}_0·{\mathbf{x}}_0$. The output of the interferometer is dispersed in a spectrometer to resolve the elastic ZLP (green) and an inelastic peak corresponding to excitation of a dipolar plasmon (orange).

Figure 2

Demonstration of inelastic interferometry. (a) Dark-field STEM image of a 60 nm gold NP on the edge of a carbon substrate. (b) Sketch of the 2GeMZI, consisting of a STEM with two gratings used as beam splitters. The first grating ($G1$) prepares electrons in a superposition of two separate paths, each of them interacting with the NP sample, with some probability of losing energy to a plasmon resonance (orange). (c),(d) The electron paths are recombined using the second grating ($G2$), which can be positioned for (c) destructive (blue borders) and (d) constructive (green borders) interference, conversely modifying the elastic and inelastic signals. Two NPs are observed in the image because of the two-spot beam configuration, with the central frame selecting the interference region (i.e., each beam passing by one side of the NP). (e),(f) For both alignment schemes, we integrate over the plasmon (yellow-shaded) and ZLP (red-shaded) regions of the energy-loss spectra plotted in (f) at every scan location to create the spectral images shown in (e). The raw spectra in (f) correspond to the dotted positions in (e). (g),(h) We simulate the experiment with an external potential $V_z(\mathbf{R})$ (g) and show that the calculated results (h) are qualitatively consistent with the experimental spectral images in (e). (i) Interference term of the loss probability, obtained as the difference of the plasmon-integrated spectral images from the two different interferometer alignments in (e) and (h) [Eq. (4)].

Figure 3

Measured and modeled ZLP and plasmon intensities as a function of total relative probe phase.