Moderate-terahertz-induced plateau expansion of high-order harmonic generation to soft X-ray region

TL;DR

This work addresses extending HHG cutoffs using readily available THz fields. By solving the time-dependent Schrödinger equation for IR+THz driving fields and analyzing electron trajectories classically and via Bohmian mechanics, it uncovers a robust fish-fin plateau structure whose overall cutoff climbs from $I_p+3.17U_p$ toward $I_p+8U_p$ and can reach $I_p+9.1U_p$ at moderate THz strengths. The mechanism is governed by long-traveling electron trajectories, with a simple analytical relation $t_e \approx T_0/(\pi\alpha)$ and $A_m \approx r_q/(2\alpha)$ explaining the saturated energy $K_{\max} \approx 8U_p$ and the cutoff behavior, consistently across atomic species and driving parameters. This demonstrates practical cutoff control with lab-scale THz fields, enabling engineered coherent EUV and soft X-ray HHG and real-time tracking of ultrafast electron motion.

Abstract

Extending the high-harmonic cutoff with experimentally accessible fields is essential for advancing tabletop coherent extreme ultraviolet (EUV) and soft X-ray sources. Although terahertz (THz) assistance offers a promising route, cutoff extension at weak, laboratory-accessible THz strengths remain poorly understood. In this report, we comprehensively investigate THz-assisted high-order harmonic generation (HHG) using time-dependent Schrödinger equation simulations supported by classical trajectory analysis and Bohmian-based quantum dynamics. By mapping the plateau evolution versus THz strength, we show that even weak THz fields can extend the cutoff, producing a pronounced ``fish-fin'' structure whose prominent rays saturate near $I_p + 8 U_p$. We trace this extension to long electron excursions spanning several optical cycles before recombination, and provide a fully consistent explanation using both classical analysis and Bohmian trajectories flow. Our findings reveal that this cutoff-extension mechanism is remarkably robust, persisting across different atomic species and remaining insensitive to variations in the driving parameters. These results demonstrate that cutoff control is achievable with laboratory-scale THz fields, offering practical guidelines for engineering coherent high-energy HHG, and providing a robust pathway for tracking ultrafast electron motion in real time.

Figures (3)

• Figure 1: (a) Comprehensive HHG spectra from the hydrogen atom in the combined IR–THz fields with varying THz strength, where the color encodes the HHG intensity. Panels [(b1)-(b4)] show detailed spectra, and Panels [(c1)-(c4)] display corresponding time–frequency profiles for specific THz fields, where $\alpha = E_T / E_0$ denotes the THz-to-IR amplitude ratio. In Panels [(b1)-(b4)], the gray curves represent the HHG spectra, whose envelope is demonstrated by the solid black curves. In Panels (c1)-(c4), the classical returning-electron kinetic-energy are shown by dotted curves. The comprehensive HHG spectra exhibits a "fish-fin" structure of the overall cutoff (white solid curve), where EUV harmonics can be generated under moderate THz fields. For $\alpha$ within $4\%-40\%$, multiplateau structure emerges due to the imbalance between adjacent attosecond bursts, where one burst group forms the first plateau [red dashed curve in panel (a)] and the other contributes to its extension.
• Figure 2: Classical simulation of returning-electron kinetic-energy at recombination (first row) and electron displacement (second and third rows) for different THz fields with $\alpha = 0$, 5%, 10%, and 40%. The corresponding Bohmian trajectories are shown in the last row where the color tone denotes the trajectory weight. For clarity, only electrons liberated within one optical cycle are presented in the first and second rows, while complete trajectories are shown in the third and last rows. In the first row, the blue and red curves represent the electron dynamics responsible for the first and second groups of attosecond bursts in Figs. \ref{['fig:compre']}[(c1)-(c4)]). In the second and third rows, gray curves denote all photoelectron trajectories, while colored curves highlight the returning ones whose colors encode their kinetic energy at recombination. The driving IR and THz fields' parameters are the same as in Fig. \ref{['fig:compre']}, except for the continuous IR envelope. Long-traveling trajectories lead to distinct burst cutoffs, thereby forming multi-plateau structures and EUV harmonics under moderate THz fields.
• Figure 3: Maximum kinetic energy (a) and maximum displacement (b) of the returning electron with different excursion time as a function of the THz field strength. The color encodes the electron’s excursion time. In Panel (a), the black dashed curve marks the overall cutoff while the black dotted curve shows the trend of the maximum kinetic energy versus the THz field. The overall cutoff exhibits the "fish-fin" structure of the comprehensive HHG. In Panel (b), the dashed black curve marks the upper limit of returning-electron displacement. As the THz strength decreases, the electron travels farther from the core, and the maximum kinetic energy decreases to a saturated value of about $8.0U_\mathrm{p}$.