Electron dynamical phase difference and high-order harmonic frequency shift: A Bohmian-trajectory perspective

Recently, we have visually elucidated the dynamics of the high-order harmonic process of an asymmetric CO molecule with a few-cycle laser pulse using the Bohmian perspective and found that only one central trajectory, BT(P), whose initial start is near the C atom, principally contributes to the high-energy part of the harmonic spectrum. The role of the other central trajectory originating near the O atom, BT(N), remains a question of interest. The present study shows the non-negligible contribution of the BT(N) to cause a frequency shift in the high harmonic spectrum when using a laser pulse with more optical cycles. Indeed, when the molecule is oriented antiparallel to the laser's electric polarization, analyzing the acceleration profiles and harmonic spectra reveals that despite smaller weight contributing to the harmonic intensity, BT(N) is significant in reproducing the harmonic spectrum structure, especially in the cutoff-energy region. The BT(N) contribution modulates the dynamical phase at the emission time, altering the dynamical phase difference between two consecutive emission bursts. This modification leads to a blueshift of harmonic orders in the cutoff region compared to the scenarios without BT(N) contribution, bringing the harmonic spectrum from the combination BTs($\mathrm{P}+\mathrm{N}$) closer to the result of directly solving the time-dependent Schr"odinger equation. This finding confirms the nonlocality of the quantum process via the Bohmian perspective that central trajectories contain all the essential dynamic information of high-order harmonic generation.