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

TL;DR

The paper addresses how the long-range Coulomb potential influences HHG electron trajectories, proposing a semi-analytical Coulomb-included TRCM that treats near-nucleus effects analytically during tunneling and numerically during rescattering. By comparing TRCM with 3D TDSE and Coulomb-modified models (MSFA, SFA), it shows that Coulomb symmetry at the tunnel exit induces a velocity $v_i$ and a lag $\tau$ between the tunneling-out time $t_0$ and the ionization time $t_i$, shifting $t_0$ earlier and enhancing short-trajectory amplitudes, with a return-time scaling law that matches TDSE. Across H and He targets and a range of laser parameters, TRCM yields HHG amplitudes and return times in close agreement with TDSE, particularly for short trajectories, while MSFA/SFA deviate in the short-trajectory regime. The work provides a clear physical picture of Coulomb effects in ionization and rescattering, delivers a quantitative framework for HHG trajectory analysis, and suggests applicability to attosecond pulse design and molecular HHG, with extensions to more complex targets. Mathematical relations such as the tunnel exit lag $\tau$, Coulomb-induced velocity $v_i$, and the Coulomb-modified momentum $\mathbf{p}'$ underpin the model’s predictive power and its alignment with first-principles simulations.

Abstract

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.

Figures (11)

• Figure 1: A sketch of the HHG process described by the SFA and the TRCM proposed here. According to the SFA, the HHG is described by a three-step process (the gray curve). That is, the electron firstly tunnels out of the barrier at the time $t_0$; then it propagates in the laser field; when the laser field changes the direction, the electron is driven to return to the nucleus, with the emission of a harmonic $\Omega$. According to the TRCM, the HHG is described by a four-step process of tunneling, response, propagation and recombination (the red curve). Specifically, when considering the Coulomb effect on tunneling, at the tunneling-out time $t_0$, the electron appears at the tunnel exit with a Coulomb-induced nonzero velocity $\textbf{v}_i$. The value of this velocity is determined by the virial theorem that puts an especial emphasis on the symmetry of the system and the direction of this velocity is opposite to the direction of tunneling (the green dotted arrow). A small period of time $\tau$ is needed for the tunneling electron to overcome this velocity. Then the tunneling electron is ionized at the time $t_i=t_0+\tau$ and begins to propagate in the laser field. The additional process between tunneling and propagation is called the response process, as the time lag $\tau$ characterizing this process reflects the basic response time of electrons within the atom to laser-induced photoionization events. When the direction of the electron velocity changes at time $t_e$ implying that a rescattering event occurs, we classically reconsider the coulomb effect in the propagation process, which accelerates the rescattering electron to return to the nucleus.
• Figure 2: Time-energy distributions of rescattering electron wave packet related to HHG, obtained by 3D TDSE for the H atom. The local maximum amplitudes are marked with black-solid circles. The energy $E_p$ is plotted in the unit of $U_p=E_0^2/(4\omega_0^2)$. The laser parameters used are $I=2\times10^{14}$W/cm$^{2}$ and $\lambda=900$ nm.
• Figure 3: Comparisons of HHG electron trajectories for the H atom, obtained with TRCM, MSFA and SFA. (a) : Tunneling-out time $t_{0}$ versus return energy $E_p$. (b): Excursion time $t_f$ versus return energy $E_p$. The laser parameters used are as in Fig. \ref{['fig:g2']}. The inset in (b) shows the difference between predictions of different models for the excursion time.
• Figure 4: Comparisons of HHG electron trajectories for the H atom, obtained with 3D-TDSE, TRCM, MSFA and SFA at $I=1.5\times10^{14}$W/cm$^{2}$ (a,b), $I=1.75\times10^{14}$W/cm$^{2}$ (c,d), and $I=2\times10^{14}$W/cm$^{2}$ (e,f). Left column: return time $t_{r}$ versus return energy $E_p$. Right Column: return time $t_{r}$ versus amplitudes of HHG electron trajectories. For comparison, in each panel in the right column, the amplitude curves of TDSE, MSFA and SFA are vertically shifted to coincide with the amplitude curves of TRCM at $t_r=1T$. The laser wavelength used is $\lambda=900$ nm. For clarity, some enlarged results are shown in the insets.
• Figure 5: Same as Fig. \ref{['fig:g4']} but for other laser parameters. The laser parameters used are $\lambda=800$ nm with $I=2.0\times10^{14}$W/cm$^{2}$ in (a,b), $\lambda=1000$ nm with $I=2.0\times10^{14}$W/cm$^{2}$ in (c,d) and $\lambda=1000$ nm with $I=1.5\times10^{14}$W/cm$^{2}$ in (e,f).