  Photonic time-crystals (PTCs) are spatially-homogeneous media whose
electromagnetic (EM) susceptibility varies periodically in time, causing
temporal reflections and refractions for any wave propagating within the
medium. The time-reflected and time-refracted waves interfere, giving rise to
Floquet modes with momentum bands separated by gaps (rather than energy bands
and gaps, as in photonic crystals). Here, we show that a free electron moving
in a PTC spontaneously emits radiation, and, when associated with momentum-gap
modes, the radiation of the electron is exponentially amplified by the
modulation of the refractive index. Moreover, under strong electron-photon
coupling, the quantum formulation reveals that the spontaneous emission into
the PTC bandgap experiences destructive quantum interference with the emission
of the electron into the PTC band modes, leading to suppression of the
interdependent emission. Free-electron physics in PTCs offers a new platform
for studying a plethora of exciting phenomena such as radiating dipoles moving
at relativistic speeds and highly efficient quantum interactions with free
electrons. Radiation emission in PTCs and its non-resonant nature hold the
promise for constructing new particle detectors with adjustable sensitivity and
highly tunable laser sources, ranging from terahertz to the x-ray regime,
drawing their power from the modulation.
