Theory of Shaping Electron Wavepackets with Light

We present the quantum theory governing the interaction between short laser pulses and relativistic free electrons and reveal intrinsic conservation laws for such quantum interactions. Through the judicious design of the amplitude and phase of a laser pulse, we propose the complete shaping of the electron temporal wavepacket and energy spectrum. To exemplify the prospects and limitations of the shaping process, we optimize laser pulses that can reduce the energy spread of an electron beam. We also find attosecond laser pulse trains that generate electron combs with record short attosecond features that push the lowest limits of electron temporal duration. We propose to utilize the quantum shaping of the electron energy spectrum for a new kind of light-matter interaction: Matching the shaped electron energy shift to a target material excitation. This way we can transfer the electron coherent state into a new coherent material excitation, allowing access to states that are forbidden with optical excitations. Our formalism generalizes all the previous theoretical models in the area of photon-induced nearfield electron microscopy (PINEM); our formalism can fit experiments in the field that cannot be modeled by the conventional PINEM theory. Altogether, our new formalism of shaping quantum electron-laser interactions in the dimension of time/energy has prospects for novel phenomena in attosecond science, for new techniques in electron microscopy, and for their potential marriage in ultrafast quantum electron optics.