  Over the last few decades, the prominent strategies for controlling
spontaneous emission has been the use of resonant or space-periodic photonic
structures. This approach, initially articulated by Purcell and later expanded
upon by Yablonovitch in the context of photonic crystals, leverages the spatial
surroundings to modify the spontaneous emission decay rate of atoms or quantum
emitters. However, the rise of time-varying photonics has compelled a
reevaluation of the spontaneous emission process within dynamically changing
environments, especially concerning photonic temporal crystals where optical
properties undergo time-periodic modulation. Here, we apply classical
light-matter interaction theory along with Floquet analysis to reveal a
substantial enhancement in the spontaneous emission decay rate at the momentum
gap frequency in photonic temporal crystals. This enhancement is attributed to
time-periodicity-induced loss and gain mechanisms, as well as the
non-orthogonality of Floquet eigenstates that are inherent to photonic temporal
crystals. Intriguingly, our findings also suggest that photonic temporal
crystals enable the spontaneous excitation of an atom from its ground state to
an excited state, accompanied by the concurrent emission of a photon.
