Analytical description of phase in high-order harmonic generation: nonadiabatic effect on single atom response and macroscopic phase matching

We propose a fully analytical expression for the phase of high-order harmonic generation (HHG) within the framework of the strong-field approximation (SFA), which clearly separates the nonadiabatic contribution from the adiabatic one. At the single-atom response level, we show that the nonadiabatic phase leads to a significant shift in the photon energy corresponding to the peak of the HHG spectrum, a phenomenon that can be attributed to the interference of different quantum trajectories. The analytical expression for the nonadiabatic phase also provides further theoretical support for reconstructing electron dynamics (excursion time) from HHG spectra generated using mixed gases. At the macroscopic level, our approach yields a fully analytical nonadiabatic expression for atomic dipole phase matching, which can accurately predict the HHG phase-matching condition and shows good agreement with both experiments and numerical SFA simulations. In particular, this expression explicitly reveals how the HHG phase depends on laser and atomic parameters. This may enable more convenient control of the HHG phase by adjusting laser parameters in future experiments aimed at attosecond pulse generation and the development of coherent high energy light sources.