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.

References

It is, however, an open problem to determine how the emitted radiation scales with the number of emitters and how this scaling is modified by phase-matching conditions.

— Colloquium: Quantum optics of intense light--matter interaction  (2510.19045 - Stammer et al., 21 Oct 2025) in Section 6.1 (Quantum field coupled to ATI electrons)

A first-principles description connecting the microscopic origin of the dipole phase with the macroscopic propagation picture, and in fact, the microscopic origin of dipole phases in high-harmonic generation by itself remains an open problem.

— A quantum optical concept of attosecond pulses: the attoquants  (2608.17854 - Varró et al., 18 Aug 2026) in Section 6, Discussion, paragraph beginning “A first-principles description…”