Scaling of ATI-induced harmonic emission with emitters and phase matching

Determine how the single-photon emission in the q-th harmonic generated by direct above-threshold ionization (dATI) scales with the number of emitters in a macroscopic medium, and ascertain how phase-matching conditions modify this scaling under realistic propagation.

Background

In the quantum-optical treatment of ATI, the direct ATI (dATI) contribution leads to a displacement of the field modes driven by the electron’s charge current in the continuum. The resulting single-photon emission probability in harmonic modes can be computed from the conditioned field state, and in single-emitter scenarios it can exceed HHG probabilities at low orders.

While these predictions are clear in the single-atom regime, extending them to macroscopic media requires understanding collective effects. In HHG, phase matching and the number of emitters critically determine the observed scaling, but for dATI-driven harmonic emission this collective scaling and its dependence on phase matching have not yet been established.