A Coulomb-included model for high-order harmonic generation from atoms

In strong laser-atom interactions, the Coulomb potential can affect the trajectories of rescattering electron in high-order harmonic generation (HHG). Here, by constructing a semi-analytical Coulomb-included model and comparing it with numerical experiments that allow for direct observation of electron trajectories, we identify the role of Coulomb potential in different processes of HHG. We show that the symmetry of the system determined by Coulomb potential plays an important role in the ionization process of HHG, inducing the tunneling-out time of electrons to shift towards earlier times. This symmetry-related effect reflects the quantum properties of atomic systems, in sharp contrast to the classical Coulomb-induced acceleration in the recombination process. In particular, compared with other strong-filed models, the scaling law of the amplitude of HHG electron trajectories predicted by this model agrees with the numerical experiments, indicating that the model developed here can be used to quantitatively describe HHG. This model can also be used to study strong-field ionization significantly influenced by rescattering.

💡 Research Summary

This paper presents a significant advancement in the theoretical modeling of high-order harmonic generation (HHG) from atoms in strong laser fields by developing a comprehensive semi-analytical model that fully incorporates the Coulomb potential’s influence on electron trajectories. The core of the work is the generalization of the Tunnel-Response-Classical-Motion (TRCM) model, originally designed for strong-field ionization, to describe the complete HHG process.

The authors begin by performing precise numerical experiments via solving the three-dimensional Time-Dependent Schrödinger Equation (TDSE) for a hydrogen atom. Using a wave-packet-tracing procedure, they extract key observables: the “return time” and the “amplitude” of the rescattering electron wave packet just before recombination. These TDSE results serve as the benchmark for evaluating theoretical models.

The study systematically compares the predictions of several models: the standard Strong-Field Approximation (SFA), which ignores the Coulomb potential; the Coulomb-Modified SFA (MSFA), which numerically integrates Newton’s equation including the Coulomb force starting from SFA initial conditions; and the newly proposed generalized TRCM model.

The generalized TRCM model offers a novel and clearer physical picture by distinguishing the Coulomb effect in the ionization phase from that in the recombination phase. For ionization, the model highlights the “near-nucleus Coulomb effect.” It posits that the Coulomb potential defines a symmetry of the system, which endows the tunneling electron with an initial velocity directed toward the nucleus at the moment it exits the tunnel. This velocity, linked to the quantum property via the virial theorem, means the electron is not immediately free. It must spend a short but crucial time lag (τ, on the order of tens of attoseconds) under the laser field’s influence to overcome this inward pull and become ionized. Consequently, the “tunneling-out time” decouples from the “ionization time” and shifts to an earlier instant compared to predictions of SFA and even MSFA. This shift is critical as it allows for electron trajectories born on the rising edge of the laser electric field and leads to a significant increase in the amplitude of short trajectories, bringing the model’s predictions much closer to TDSE results.

In contrast, during the recombination phase, as the electron is driven back toward the nucleus, the Coulomb potential acts as a classical accelerating force, pulling the electron in and causing an earlier return time compared to SFA—an effect also captured by MSFA.

The most compelling validation of the TRCM model comes from its prediction of the “scaling law” governing how the amplitude of HHG electron trajectories (for both short and long paths) varies with their return time. While the MSFA’s scaling law shows notable deviations from TDSE results under certain laser parameters, the TRCM model’s predictions show excellent quantitative agreement with TDSE across a wide parameter range. Since this scaling law reflects the fundamental dynamics of the HHG process, this agreement strongly indicates that the generalized TRCM model is a robust and accurate tool for the quantitative description of HHG. Furthermore, the model’s foundation suggests it can be effectively applied to study other strong-field phenomena, such as ionization processes dominated by rescattering events.