Highly accurate semiclassical strong-field Herman-Kluk propagator method for high-harmonic generation

TL;DR

This work extends the semiclassical strong-field Herman–Kluk (SFHK) propagator to high-order harmonic generation (HHG), achieving TDSE-like accuracy by birth of continuum wave packets via the strong-field approximation and propagation under the full atomic potential with the Herman–Kluk propagator. The method retains the full pre-exponential HK factor, handles all relevant classical trajectories, and is highly parallelizable, enabling efficient 3D HHG calculations. Demonstrations on atomic hydrogen and argon show excellent agreement for yields and phases, reproduce the Cooper minimum, and provide time–frequency insights into recombination dynamics, including precise recombination times. The study also surveys the use of ADK-type tunneling within SFHK and discusses extensions to molecular targets, suggesting SFHK as a rigorous and scalable tool for benchmarking strong-field ionization theories.

Abstract

We extend our recently developed semiclassical strong-field Herman-Kluk propagator (SFHK) method to calculate high-order harmonic generation (HHG) for atoms in intense lasers. We show that our method, based on a combination of the Herman-Kluk propagator and the strong-field approximation, can provide highly accurate results for both HHG yield and phase, nearly identical to those from the exact numerical solutions of the time-dependent Schrödinger equation. We provide detailed analyses of our method and its applications to the HHG process, particularly the recombination time. The main computational task in this method is to solve the classical Newton equations for the active electron in the combined atomic potential and laser-electron interaction. The motion of the centers of the electron wave packets, modeled by coherent states, is governed by independent classical trajectories so that the computation can therefore be parallelized very efficiently.

Figures (8)

• Figure 1: Comparison of HHG yield (a) and phase (b), vs emitted photon energy, calculated within the SFHK and TDSE for atomic hydrogen. The vertical axis in (a) is on a logarithmic scale. A three-cycle laser pulse with the wavelength of 1,600-nm, intensity of $10^{14}$ W/cm$^2$, and CEP $\phi=0$ was used in the calculations.
• Figure 2: The Gabor time-frequency analysis of the induced dipole velocity calculated with the TDSE (a) and the SFHK (b). The color map is on a logarithmic scale. The electric field of the laser pulse is also shown as the dashed line. The laser parameters are the same as in Fig. \ref{['fig:H1600']}.
• Figure 3: (a) Comparison of the emission time calculated using the Gabor transform of the dipole velocity from the SFHK and the TDSE. See text for more detail. The electric field of the laser pulse is also shown as the dotted line. (b) The zoom-in of (a) in the time range between 160 as to 320 as. The laser parameters are the same as in Fig. \ref{['fig:H1600']}.
• Figure 4: (a) and (b): same as in Fig. \ref{['fig:H1600']} but for the wavelength of 1,200-nm. (c) and (d): same as in (a) and (b), respectively, but for a six-cycle laser pulse. (e) and (f): same as (a) and (b) but for the wavelength of 800-nm. The vertical axis in (a), (c), and (e) is on a logarithmic scale.
• Figure 5: (a), (b), and (c): Partial-wave analysis of the trajectories returning to within $r < 3$ a.u. from the core, for $s$-wave, $p$-wave, and $d$-wave, respectively, for atomic hydrogen case. See text for more detail. (d) and (e): comparison of HHG yield and phase calculated with the TDSE with the SFHK, but only $p$-wave trajectories that come close to within $r < 3$ a.u. from the core were included in the SFHK. The vertical axis in (d) is on a logarithmic scale. The laser parameters are the same as in Fig. \ref{['fig:H1200']}(a) and (b).