Efficient time-sampling method in Coulomb-corrected strong-field approximation

One of the main goals of strong-field physics is to understand the complex structures formed in the momentum plane of the photoelectron. For this purpose, different semiclassical methods have been developed to seek an intuitive picture of the underlying mechanism. The most popular ones are the quantum trajectory Monte Carlo (QTMC) method and the Coulomb-corrected strong-field approximation (CCSFA), both of which take the classical action into consideration and can describe the interference effect. The CCSFA is more widely applicable in a large range of laser parameters due to its nonadiabatic nature in treating the initial tunneling dynamics. However, the CCSFA is much more time consuming than the QTMC method because of the numerical solution to the saddle-point equations. In the present work, we present a time-sampling method to overcome this disadvantage. Our method is as efficient as the fast QTMC method and as accurate as the original treatment in CCSFA. The performance of our method is verified by comparing the results of these methods with that of the exact solution to the time-dependent Schrödinger equation.

Figure 1

The top panels show the 2D photoelectron momentum distributions of hydrogen atom, ionized by a 4-cycle linearly polarized laser pulse at wavelength of 800 nm and peak intensity of $1\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, with a ${cos}^2$ envelope: (a) direct numerical integration from Eq. (7) and [(b) and (c)] our time-sampling method without and with the Jacobian matrix respectively. The low panels show the normalized momentum spectra for $p_z$ at $p_x=0,0.02,0.1$ respectively in (d), (e), and (f). The red solid line is extracted from the exact result in panel (a), and the green long-dashed line is extracted from our method in panel (c). Note that the Coulomb potential is temporally neglected in these calculations.

Figure 2

The 2D photoelectron momentum distributions of hydrogen atom, after including the Coulomb potential. The laser parameters are the same as those in Fig. 1. The results are shown from: (a) TDSE method, (b) QTMC method, (c) ordinary CCSFA method, and (d) time-sampling CCSFA method. The area within the black box is zoomed in the left-top corner for the finer structures. See text for more detailed discussions.

Figure 3

The trajectories (left panels) and the real part of the complex ionization time (right panels) of the sample electrons that marked by a small circle in the zoomed area of Figs. 2–2. The electric field is shown in black line in right panels. The sample electrons are selected with the final momentum around $p_x=-0.7,p_y=0.1$. The top panels show the results calculated by the QTMC method and the bottom panels by our time-sampling method.