Boundary memory in ultrafast field emission: a two-time quantum statistical theory

Ultrafast field emission is usually modelled by evaluating the stationary tunnelling current at the instantaneous driving field. We show that this fails whenever the emitting surface has electronic structure near the Fermi level: the vacuum escape channel reacts back on the occupied boundary spectrum that feeds it, and because that spectrum relaxes on a finite timescale the emitted current carries memory of the field's recent history. Formulating the driven emission problem on the Keldysh contour and propagating the boundary Green functions with a material specific surface spectrum, we find two distinct kinds of memory: a traversal memory that filters and delays the carrier when the driving frequency approaches the tunnelling rate, and a spectral relaxation memory that suppresses and reshapes the current when a pulse drives a thin barrier emitter through the stationary overcoupling turnover. The second can dominate even when the barrier traversal is quasistatic, and predicts a current that falls while the instantaneous transparency still rises. Which memory governs a surface is set by its barrier thickness and its coupling to the vacuum, so the ultrafast waveform distinguishes emitters that the instantaneous theory treats alike.