Electron microscopy and diffraction have long been used to study the structure of materials. A recent development is the achievement of ultrafast time resolution using high-repetition rate single electron sources and single-shot many-electron sources. Achieving large coherence lengths under these conditions is challenging due to intrinsic heating mechanisms including inter-particle Coulombic forces. Near-threshold, pulsed photoionization of atoms trapped and cooled with lasers is being investigated as a means to produce intense electron pulses with large intrinsic coherence length due to a low electron temperature. Photo-ionizaton sources are, of course, not an entirely new idea but recent implementations using trapped atoms offer new possibilities. Several properties of electron beams extracted from such ultracold sources have already been investigated, showing in particular that temperatures of a few Kelvin and emittances in the nanometer-radian range can be achieved with tens of thousands electrons per pulse for nanosecond pulse lengths. This can be combined with delicate shaping of the charge distribution to counteract emittance growth due to non-linear self fields. Using ultrafast lasers for the ionization step, shorter electron pulses can be achieved, but an interesting question is whether the concomitant spread in photon energy still allows for low electron temperature. Recent experiments have shown that few-Kelvin temperatures combined with picosecond pulse lengths are indeed possible. It turns out that this can be understood with a classical model for electron trajectories resulting from photoionization. In addition, experimentally observed polarization effects are in general agreement with model results. --------------------------------------------------------------------------------
Time-resolved x-ray diffraction (TR-XRD) and ultrafast electron diffraction (TR-UED) are emerging tools for probing ultrafast quantum dynamics. From a theoretical perspective, they are commonly described within different frameworks and modeled using distinct approximations. Here, we present a unified quantum-field-based description of ultrafast diffraction imaging that permits consistent consideration of TR-XRD and TR-UED within a common theoretical formalism. Our approach elucidates the correspondence between TR-XRD and TR-UED and allows their similarities and differences to be systematically disentangled. The developed formalism is sufficiently general to consistently and straightforwardly incorporate additional physical effects of interest, such as relativistic charge-current and current-current couplings. We apply our approach to simulate diffraction measurements of laser-driven electron dynamics in graphene, demonstrating the unique capabilities of diffraction imaging to unravel intricate quantum processes in matter.