A spatial cavity confines light by reflection — mirrors at both ends create standing waves at discrete frequencies. The confinement produces the physics of cavity QED: enhanced atom-light coupling, Rabi oscillations, modified spontaneous emission. The cavity is a structure in space.
A photonic temporal crystal modulates the optical properties of a medium periodically in time rather than space. Where spatial periodicity creates frequency gaps (bands of forbidden frequencies), temporal periodicity creates momentum gaps (bands of forbidden momenta). The mathematics is analogous; the physics is rotated.
The paper develops the quantum theory of temporal cavities and finds that the analogy breaks at a critical point. The classical momentum gap — a band of forbidden momenta — corresponds quantum-mechanically to a localization-delocalization phase transition in a Floquet-photonic synthetic lattice. Inside the gap, photonic states are delocalized across the synthetic lattice. Outside, they are localized. The transition is sharp.
An atom embedded in a temporal cavity shows irreversible Rabi oscillation decay driven by a single frequency mode. In a spatial cavity, irreversible decay requires coupling to a continuum of modes — a single mode produces persistent Rabi oscillations. The temporal cavity achieves what the spatial cavity cannot: decoherence from a single mode, because the Floquet structure provides an effective continuum through the synthetic lattice dimension.
The structural point: swapping space for time in optical confinement does not merely translate the physics — it produces qualitatively new phenomena. The temporal dimension provides a synthetic degree of freedom that spatial cavities lack. Time is not just another direction; it opens channels that space does not.