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We introduce a framework for the preparation, coherent manipulation andcharacterization of free-electron quantum states, experimentally demonstratingattosecond pulse trains for electron microscopy. Specifically, we employphase-locked single-color and two-color optical fields to coherently controlthe electron wave function along the beam direction. We establish a new variantof quantum state tomography - "SQUIRRELS" - to reconstruct the density matricesof free-electron ensembles and their attosecond temporal structure. The abilityto tailor and quantitatively map electron quantum states will promote thenanoscale study of electron-matter entanglement and the development of newforms of ultrafast electron microscopy and spectroscopy down to the attosecondregime.

Attosecond Electron Pulse Trains and Quantum State Reconstruction in Ultrafast Transmission Electron Microscopy