Modulation of high-harmonic generation by coherent two-phonon dynamical coupling

We study delay-dependent high-harmonic generation in a two-dimensional (2D) material modulated by two coherent phonons. In our first-principles calculations, two phonon modes are simultaneously initiated by displacive launch and a time-delayed probe is used to drive the HHG. We observe two robust enhancement windows ($\sim$13 and $\sim$50 fs) in the HHG plateau yield. Notably, these enhancements do not coincide with displacive extrema of either single composed phonon mode. Based on a minimal harmonic model, we describe this behavior involving the delay ordering and phonon-phase sensitivity. Through analysis of the phonon-enhanced excited-electron number and the k-selective photocarrier injection concentration, we reveal a complex phonon coupling effect: the out-of-plane ZO mode predominantly modulates the band gap, establishing the phonon-driven carrier-excitation baseline and the characteristic momentum-space distribution, whereas LO-induced valley reshaping further enhances carrier accumulation near M and strengthens plateau emission at the optimal delays. These findings establish two-phonon mixing as an effective control knob for optimizing HHG in 2D materials.