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Journal of Physics B Atomic Molecular and Optical Physics

Revisiting the time-dependent ionization process through the Bohmian-mechanics method

Yang Song, Yujun Yang, Fu-Ming Guo, Suyu Li

Published Feb 27, 2017• 11 citations
Publisher

Abstract

Abstract The ionization process of a hydrogen-like atom in the intense laser field is studied by the Bohmian-mechanics method. We propose a criterion for atomic ionization based on the total energy of a Bohmian particle (BP): when it is larger than or equal to zero, ionization occurs, when it is lower than zero, no ionization occurs. We find that the time-dependent ionization probabilities calculated by the Bohmian-mechanics and projection methods are almost the same under different laser parameters. Two typical Bohmian trajectories are selected to analyze the ionization process. The consistency between the time-dependent ionization probabilities calculated by the two methods is quantitatively analyzed from the aspect of the probability density of the wave-function.

DOI
10.1088/1361-6455/aa630d

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Abstract The ionization process of a hydrogen-like atom in the intense laser field is studied by the Bohmian-mechanics method. We propose a criterion for atomic ionization based on the total energy of a Bohmian particle (BP): when it is larger than or equal to zero, ionization occurs, when it is lower than zero, no ionization occurs. We find that the time-dependent ionization probabilities calculated by the Bohmian-mechanics and projection methods are almost the same under different laser parameters. Two typical Bohmian trajectories are selected to analyze the ionization process. The consistency between the time-dependent ionization probabilities calculated by the two methods is quantitatively analyzed from the aspect of the probability density of the wave-function.
The ionization dynamics of two-electron atom in an intense laser field is studied by the Bohmian mechanics (BM) theory, and the xenon atomic potential function is used as a model. The single ionization process and double ionization process are calculated by the BM theory and their results are in good agreement with those calculated by numerically solving the time-dependent Schrödinger equation. The analyses of the types, trajectories, and forces of Bohmian particles (BPs) undergoing the single and double ionizations indicate that the re-collision process accounts for a considerable proportion in the singly ionized cases. Furthermore, the analysis of the work done by the external force acting on the BPs shows that the quantum force plays an important role in the re-collision process. This work is helpful in understanding the ionization of two-electron atom in an intense laser field.
Resonance enhanced two-photon ionization process of hydrogen atom via the resonant laser pulse is studied by Bohmian mechanics (BM) method. By analyzing the trajectories and energies of Bohmian particles (BPs), we find that under the action of high frequency and low intensity multi-circle resonant laser pulses, the ionized BPs first absorb one photon completing the excitation, and then absorb another photon, completing the ionization after staying in the first excited state for a period of time. The analysis of work done by the forces shows that the electric field force and quantum force play a major role in the whole ionization process. At the excitation moment and in the excitation-ionization process, the effect of the quantum force is greater than that of the electric field force. Finally, we discuss the principle of work and energy for BPs, and find that the electric field force and quantum force are non-conservative forces whose work is equal to the increment of mechanical energy of the system. In addition, it is proved that the quantum potential energy actually comes from the kinetic energy of the system and the increment of kinetic energy is equal to that of the kinetic energy of the system.
In this paper, Bohmian mechanics is applied to intense laser–atom physics. The motion of an atomic electron in an intense laser field is obtained from the Bohm–Newton equation. We find that the quantum potential that dominates the quantum effect of a physical system becomes negligible as the electron is driven far from the parent ion by the intense laser field, i.e. the behavior of the electron smoothly tends towards classical soon after the electron is ionized. Our numerical calculations present direct positive evidence for semiclassical trajectory methods in intense laser–atom physics where the motion of the ionized electron is treated by classical mechanics, while quantum mechanics is needed before the ionization.
The light-emission process by the recombination of an ionized electron with its parent ion is investigated by the Bohmian trajectory scheme, and the efficiency of light emission calculated by 20 Bohmian trajectories is qualitatively identical to that obtained by numerically solving the time-dependent Schr\"odinger equation. It is found that the efficiency of light emission by recombination is determined by the oscillation amplitude of the Bohmian particle's Coulomb acceleration. Along with the decrease of the energy of the incident Bohmian particle, the deepening of the ion's potential well, and the increase of the product factor of the bound-state and continuum-state populations, the oscillation amplitude of the particle's Coulomb acceleration increases.
The exact factorization of an electron-nuclear wave function [A. Abedi, N. T. Maitra, and E. K. U. Gross, Phys. Rev. Lett. 105, 123002 (2010)] allows us to define the rigorous nuclear time-dependent Schr\"odinger equation (TDSE) with a time-dependent potential-energy surface (TDPES) that fully accounts for the coupling to the electronic motion and drives the nuclear wave-packet dynamics. Here, we study whether the propagation of multiple classical trajectories can reproduce the quantum nuclear motion in strong-field processes when their motions are governed by the quantum Hamilton-Jacobi equation derived by applying Bohmian mechanics to this exact nuclear TDSE. We demonstrate that multiple classical trajectories propagated by the force from the gradient of the exact TDPES plus the Bohmian quantum potential can reproduce the strong-field dissociation dynamics of a one-dimensional model of the H2+{{\mathrm{H}}_{2}}^{+}H2​+ molecule. Our results show that the force from the Bohmian quantum potential plays a non-negligible role in yielding quantum nuclear dynamics in the strong-field process studied here, where ionization and/or splitting of nuclear probability density occurs.
Recent attoclock experiments and theoretical studies regarding the strong-field ionization of atoms by few-cycle infrared pulses revealed features that have attracted much attention. Here we investigate tunneling ionization and the dynamics of the electron probability using Bohmian mechanics. We consider a one-dimensional problem to illustrate the underlying mechanisms of the ionization process. It is revealed that in the major part of the below-the-barrier ionization regime, in an intense and short infrared pulse, the electron does not tunnel through the entire barrier, but rather starts already from the classically forbidden region. Moreover, we highlight the correspondence between the probability of locating the electron at a particular initial position and its asymptotic momentum. Bohmian mechanics also provides a natural definition of mean tunneling time and exit position, taking account of the time dependence of the barrier. Finally, we find that the electron can exit the barrier with significant kinetic energy, thereby corroborating the results of a recent study [N. Camus et al., Phys. Rev. Lett. 119, 023201 (2017)].