Entangled trajectories during ionization of an H atom driven by n-cycle laser pulse

Highlights

• For a two-cycle laser pulse, the entangled trajectory is simple compared with that for a four-cycle laser pulse.

• For two-cycle or four-cycle pulses, the influence of the quantum force is more distinct than that of the classical force.

• The final kinetic energy of the entangled trajectories is slightly higher than that obtained by the classical calculation.

• The convergence of the trajectory equations is checked.

Abstract

We study the ionization of an H atom in a linearly polarized laser field with different optical cycle numbers by calculating the entangled trajectories in phase space. The results indicate that, for a two-cycle laser pulse, the entangled trajectory is simple owing to the simple laser electric distribution. As the number of optical cycles increases, the complexity of the laser electric field distribution, and subsequently, that of the entangled trajectories increase. From these entangled trajectories, re-scattering ionization can be observed. Further, we investigate the effect of quantum force on trajectories by comparing them with classical trajectories. We find that for few-cycle laser pulses ($n_p=2$ and 4), the effect of the quantum force is more distinct than classical behavior, whereas for longer laser pulses ($n_p=6$ and 8), it is quite similar to the classical behavior. Because of the quantum force, the final kinetic energy is slightly higher than that obtained by classical calculation. The different initial positions have some influence on the individual trajectories, but the individual trajectories still keep similar configuration to the mean trajectories.

Introduction

Interaction between atoms or molecules and few-cycle laser pulses can result in many fundamental strong-field physics phenomena such as above-threshold ionization (ATI), high harmonic generation (HHG), and non-sequential double ionization (NSDI) [1], [2], [3], [4], [5], [6], [7], [8]. Ionized electrons from atoms and molecules in a few-cycle pulsed laser field are emitted in direct ionization or are pulled back by the laser field when the laser electric field changes direction. The ionized electrons can carry information about the laser field and the electronic structure of the atoms or molecules. For example, Milošević et al. observed and measured the carrier-envelope (CE) phase of a laser pulse through the strong left-right asymmetry of the emitted electrons[9], [10]. Paulus et al. investigated the ionization of an H atom exposed to an intense few-cycle laser pulse by calculating the left-right asymmetry of the photoelectron momentum distributions, and they found an asymmetry parameter as a function of laser intensity for a particular CE phase range [11]. The results reported by Morishita et al. implied that existing few-cycle infrared lasers can be implemented for the ultrafast imaging of transient molecules with a temporal resolution of a few femtoseconds [12]. Huismans et al. formulated a new method to record the underlying electron dynamics on a sub-laser-cycle timescale, enabling photoelectron spectroscopy with a temporal resolution using the holographic structures of the ionization of a Xe atom [13]. Few-cycle laser pulses are characterized by parameters such as the frequency and intensity of the laser electric field, CE phase, and number of optical cycles. The effect of frequency and intensity on ionization has been reported. The actual shape of a few-cycle pulse crucially depends on the number of optical cycles, and so does the physical processes. In our earlier studies [14], [15], we investigated the ionization behaviors of an H atom in a few-cycle pulse with different optical cycle numbers and found that photoelectron angular distributions exhibit energy-dependent phenomena for different ATI peaks, and ionization behaviors of the atom extend to those of an atom in an infinitely monochromatic plane wave when the optical cycle numbers are very large.

Theoretical calculations for interactions between atoms or molecules and intense few-cycle laser pulses are typically carried out using different methods. The main theoretical approaches can be divided into two groups: the first kind is the direct integration of the time-dependent Schrödinger equation (TDSE), or the Bohmian formulation basing on the TDSE [16], and the second kind is analytical theory, such as the Keldysh–Faisal–Reiss (KFR) model [17], [18], [19], the analytical R-matrix method [20], [21]. In addition to the above methods, many people have used the Wigner function to investigate the ionization of atoms or molecules in strong laser field [22], [23], [24]. The Wigner function [25], [26], [27], [28] is a quasiprobability function in phase space that allows one to study position-momentum correlations. These correlations give a physical interpretation of the emergence of the above-threshold-ionization (ATI) energy spectrum. The Wigner function enables us to show the dependence of ionization probability on time and explicitly demonstrates the transition of the ionization process. In this study, we follow the approach of our recent work [22], however, we do not seek further detailed improvements on this approach. We perform comprehensive calculations of the entangled trajectories of the electrons which have been ionized irrespective of the ionization mechanism (tunneling ionization or over barrier ionization). From these trajectories, we can investigate the effect of the optical cycle number on entangled trajectories. Further, we demonstrate the effect of quantum force on the trajectories by comparing them with classical trajectories.

This paper is arranged as follows: we introduce the Wigner function used in this paper in Sec. 2 The results and discussions are in Sec. 3. In Sec. 4 we conclude.