Synthetic gain for electron-beam spectroscopy.

Electron-beam microscopy and spectroscopy featuring atomic-scale spatial resolution have become essential tools used daily in almost all branches of nanoscale science and technology. As a natural supercontinuum source of light, free electrons couple with phonons, plasmons, electron-hole pairs, inter- and intra-band transitions, and inner-shell ionization. The multiple excitations, intertwined with the intricate nature of nanostructured samples, present significant challenges in isolating specific spectral characteristics amidst complex experimental backgrounds. Here we introduce the approach of synthetic complex frequency waves to mitigate these challenges in free-electron-light interaction. The complex frequency waves, created through causality-informed coherent superposition of real-frequency waves induced by free electrons, offer virtual gain to offset material losses. This amplifies and enhances spectral features, as confirmed by our electron energy loss and cathodoluminescence measurements on multilayer membranes, suspended nanoparticles, and film-coupled nanostructures. Strikingly, we reveal that our approach can retrieve resonance excitation completely buried underneath the zero-loss peak, substantially enhance the quality of hyperspectral imaging, and resolve entangled multiple-photon-electron events in their quantum interaction. Our findings indicate the versatile utility of complex frequency waves in various electron-beam spectroscopy and their promising diagnostic capabilities in free-electron quantum optics.