The coherent control of ultrafast electron beams has recently attracted much attention for its potential application in ultrashort (attosecond) electron sources, as well as electron imaging and spectroscopy. While semi-infinite light beams have been used for the temporal streaking and compression of electron pulses21,32,33,34, here we demonstrate the simultaneous quantized exchange of energy and transverse momentum between electrons and light, which is the dominant mechanism at optical frequencies and above, and we provide a direct measurement of the strength of this quantum coherent interaction for controlling the electron energy-momentum distribution. In our experiments, we synthesize a semi-infinite temporally modulated field distribution (obtained by a sequence of two mutually phase-locked light pulses impinging on a mirror) to demonstrate coherent modulation of the electron wave function. A schematic representation of such modulation is shown in Fig. 5e, where snapshots of the strong electron density redistribution in both energy and momentum, as observed experimentally and calculated theoretically, are presented for different values of the optical phase shift of the synthesized optical field distribution. This approach allows us to develop additional capabilities of coherent control of free-electrons beyond similar configurations adopted so far, where the electron wave function modulation is determined by the light interaction at different spatial positions along the electron pathway20,21,32,33,34. In our scheme, the adoption of two temporally separated semi-infinite light fields on one flat and homogeneous thin layer allows us to employ a simpler experimental geometry and shift the two interactions temporally instead of spatially, thus taking full advantage of the intrinsic longitudinal coherence of the single-electron wave function.
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