Universal Free-Electron-Sideband Control

Advances in free-electron quantum optics are establishing free electrons as a versatile quantum resource. Photon-induced near-field electron microscopy (PINEM) interactions and free drift provide elementary operations for shaping the longitudinal electron wavefunction, yet their full control capabilities remain unexplored. Here, we prove that a finite sequence of alternating monochromatic PINEM interactions and free drifts can approximate any prescribed inner-product-preserving map from a finite-dimensional input subspace into the full ideal electron-sideband Hilbert space to arbitrary accuracy. We therefore call the protocol PINEMAD (PINEM And Drift). This framework encompasses state preparation, unitary gates, and isometric encoding. We further show numerically that parameter optimization enables high-fidelity state preparation at finite electron energy spreads, using experimentally accessible coupling strengths and tabletop-scale total drift lengths. These results establish electron energy sidebands as a programmable quantum resource.