Driving-frequency scaling of high-harmonic quantum paths

We have developed an analytical theory explaining how the single-atom efficiency of high-harmonic generation scales with laser frequency, and verified this by numerically solving the time-dependent Schrödinger equation in three spatial dimensions. According to our saddle-point analysis of quantum paths, the imaginary part of the action has a significant impact on the scaling law. Furthermore, we found that the scaling law depends on the analytical properties of the ground–continuum transition matrix element. Our analysis elucidates how the relative contributions of different quantum orbits and their relative phases vary with the driving laser frequency and how the resulting quantum-path interferences in high-harmonic spectra can be controlled with an attosecond accuracy.