  We propose a synthetic dimension for ultrafast free electrons based on the
discrete energy sidebands in photon-induced near-field electron microscopy
(PINEM). The PINEM synthetic dimension can offer a powerful way to coherently
shape or modulate free-electron wavefunctions in systems such as ultrafast
transmission electron microscopes, dielectric laser accelerators, and quantum
free-electron lasers. As examples of this paradigm, we demonstrate Bloch
oscillations, diffraction management, and linear optics effects such as Talbot
self-imaging in the PINEM lattice. These effects show the potential of PINEM
synthetic dimensions as a novel quantum simulation platform.
