Investigation of the Spatio-Temporal Characteristics of High-Order Harmonic Generation Using a Bohmian Trajectory Scheme

Abstract

High-order harmonic generation of atoms irradiated by an ultrashort laser pulse was calculated by numerically solving the time-dependent Schrödinger equation and the Bohmian trajectory scheme. The harmonic spectra with the two schemes are quantitatively consistent. Using the wavelet behavior of the Bohmian trajectory, the spatio-temporal features of harmonic emission from different energy regions are analyzed. It is found that the spatio-temporal distribution of the harmonic well revealed the physical mechanism of harmonic generation. This method will contribute to the understanding of harmonic emission mechanisms in complex systems, which include many atoms.

1. Introduction

High-order harmonic generation (HHG) can be created by ultrafast intense laser fields interacting with gas atoms [1,2,3,4,5], molecules [6,7,8,9,10,11], and solids [12,13,14,15]. As harmonic energy rises, the intensity of harmonic emission develops a distinct “platform” structure. As a result, desktop coherent XUV and soft X-ray laser sources could be obtained by HHG. High-order harmonic spectrum has a wide range of energy, making it possible to produce attosecond (10⁻¹⁸ s) ultrashort pulses [16,17,18,19]. One can achieve dynamic detection of electrons in atoms and molecules by combining the few-cycle infrared pulse with the ultrashort attosecond pulse [20,21,22,23,24]. Furthermore, since the recombination of ionized electrons and parent ions leads to HHG, the harmonic spectrum carries the structural information of atomic and molecular targets. As a result, HHG is used to perform electron orbital “imaging” of atoms and molecules [25,26,27,28].

Typically, a semi-classical three-step model is adopted by HHG [29,30,31]. First, the bound electrons are released from the barrier created by the atomic potential and the laser electric field, and tunnel ionization in the laser electric field. The laser electric field then causes the ionized electrons to accelerate like classical charged particles. Finally, under the influence of the laser electric field, certain electrons have the opportunity to recover to the ground state of the atom, emitting high-energy photons. In the case of a single atom, the cutoff energy of the atom’s high-order harmonic may be accurately predicted by this model.

Despite the fact that this model can provide accurate predictions about the high-order harmonic cutoff energy, it is unable to explain the harmonic efficiency and structural details of its spectrum. This is due to the fact that the model relies on a classical description of electrons rather than treating it as a wave packet, making it unable to collect quantum information such as transition probabilities. In order to understand the harmonic emission process, quantitative rescattering (QRS) schemes and strong field approximation theory have been developed [32,33,34]. However, these schemes usually contain only one bound state of the system, and the system’s excitation and ionization calculations are not precise enough, which has an impact on HHG. If we employed the longer wavelength laser, the ionization of atoms illuminated by this laser not only occurs in the ground state but also in the excited state. More crucially, this theory considers ionized electrons to be free electrons traveling in a laser field, ignoring the influence of coulomb potential. The phase of harmonic emission is also affected by the change in coulomb potential, which in turn influences the intensity and interference structure of harmonics [35,36,37,38]. HHG obtained in the experiment is a coherent superposition of all atomic harmonics in the gas target so that the precision of the harmonic phase of a single atom will affect the accuracy of the harmonic calculation as a whole [39].

It has recently been suggested to use the Bohmian trajectory (BT) scheme based on the wave packet technique to examine the generation process of HHG. This method is based on the numerical solution of the time-dependent Schrödinger equation (TDSE). The study of the harmonic generating process is followed by the computation of the Bohmian particle motion behavior and associated harmonic emission [40,41,42,43,44,45]. Song et al. demonstrate that the harmonic structure may be qualitatively replicated by a single Bohmian trajectory using this approach. Additionally, they contrast the harmonics of various motion types’ Bohmian trajectories [42,43,44,45,46]. By employing two Bohmian trajectories positioned at the center of the nucleus, Wang et al. investigated the coherent minimum structure of harmonic production from molecular ions and described this process [46,47]. Li et al. continued to analyze the behavior of several Bohmian trajectories and compared the harmonic production of the two different types of trajectories.

By examining the partial trajectory of the peak location of the ground state wave packet in the studies mentioned above, the process of harmonic generation may be qualitatively depicted. Tens of thousands of Bohmian trajectories are needed to achieve quantitative and accurate harmonic calculation [45]. Using these trajectories to examine the space-time behavior of harmonic emission is highly challenging. With the goal of producing a precise and quantitative harmonic spectrum, this research proposes to investigate the spatial and temporal behavior of harmonic emission using a limited number of weighted Bohmian trajectories. It is discovered that the space-time emission behavior of various order harmonics exhibits significant variances. The spatio-temporal behavior of atomic high-order harmonic emission is studied by the weighted Bohmian trajectory scheme. The results show that the lower harmonics below the threshold and the higher harmonics in the platform region have significant differences in the symmetry of the spatial position distribution of radiation. The higher-order harmonic emission is located in the nuclear region and symmetrically distributed, while the lower-order harmonic emission is located far away from the nucleus and only on one side of the nucleus, no longer symmetrically distributed about the nucleus. The theory underlying the study of electron behavior at attosecond time scales and angstorm space scales is predicted to be helped by this discovery, which is also expected to increase harmonic efficiency. The following arrangement is used: The theoretical framework employed in the research is presented in the second part. The third part presents the results, along with an analysis and discussion based on them. The final part is a summary. Unless specified, atomic units are used in this article.

2. Materials and Methods

In order to study the space-time behavior of HHG, we should calculate the Bohmian trajectory, which requires the wave function of the system. It can be numerically solved TDSE to obtain the time-dependent wave function of the system [48]

$$i\frac{\partial }{\partial t}\psi \left(x,t\right)=\left[-\frac{1}{2}\frac{{\partial }^2}{\partial x^2}+V_a\left(x\right)+x·E\left(t\right)\right]\psi \left(x,t\right)$$

(1)

The potential function is:

$$V\left(x\right)=-\frac{q}{\sqrt{x^2+a}}$$

(2)

here, we choose to use the soft nuclear potential. The soft nuclear parameters are q = 1 and a = 0.4826, respectively. The ground state energy of this system is −0.9 (the ground state energy of a helium atom).

The incident laser electric field is

$$E\left(t\right)=F0f \left(t\right)sin\left(\omega t\right)$$

(3)

f (t) is the Gaussian pulse envelope.

The time-dependent wave function of the system is obtained by solving Equation (1) using the split operator [49,50] scheme. In the method, the time-dependent wave function

Y(x, t) is calculated by the wave function Y(x, t − dt) at the previous instant. The time-dependent evaluation operator is the second-order symmetrized splitting operator. The time-dependent Hamilton is divided into two parts: the space and the momentum. The action of the momentum operation on the wavefunction is applied by the Fourier transform from the spatial function [2]. The trajectory of Bohmian particles can be obtained from the wave function of the system. The velocity of Bohmian particles can be expressed as:·

$$vk \left(t\right)=Im \left[\frac{1}{\psi \left(x, t\right)}\frac{\partial }{\partial x} \psi \left(x, t\right) ,x=xk \left(t\right)\right] \left(k=1, 2, . . . , N\right)$$

(4)

The trajectory information of the Bohmian particle can be calculated by using its velocity information $x^k\left(t\right)=x^k\left(t=0\right)+{\int }_0^t{v}_0^k\left(t^{\prime }d{t}^{\prime }\right)$. Using the initial wave function information of the system, the weights of different Bohmian particles are given $w^k{\left|\psi \left(x^k\left(t\right),0\right)\right|}^2$, then the time-dependent dipole moment of the system is calculated as:

$$x\left(t\right)={\displaystyle \sum }_{k=1}^N{w}_i^k{x}_i^k\left(t\right)$$

(5)

In order to obtain the spatial and temporal information of harmonic frequency, we use wavelet transform to time-dependent dipole moment. The Morlet wavelet can be expressed as:

$$A_{x_i^k}\left(t_0,x_i^k\right)={{\displaystyle \int }}_{\mathit{ti}}^{t\tau }{w}_{t_0,\omega }\left(t\right)x_i^k\left(t\right)w_i^{k}dt$$

(6)

$w_{t_0,\omega }\left(t\right)$ is the kernel of wavelet transform:

$$w_{t_0,\omega }\left(t\right)=\sqrt{\omega }W\left[\omega \left(t-t_0\right)\right]$$

(7)

where $W\left(s\right)=\frac{1}{\sqrt{\tau }}{e}^{is}{e}^{-\frac{s^2}{2{\tau }^2}}$.

In order to compare the harmonic spectrum obtained by the Bohmian trajectory, we also use the TDSE method to obtain the time-dependent wave function of the system and calculate the time-dependent dipole moment of the system. The time-dependent dipole moment in length form and the dipole moment in acceleration form are:

$$X_{TDSE}\left(t\right)=-{{\displaystyle \int }}_{-\infty }^{\infty }{\psi }^{*}\left(x,t\right)x\psi \left(x,t\right)dx$$

(8)

$$a_{TDSE}\left(t\right)=-{{\displaystyle \int }}_{-\infty }^{\infty }{\psi }^{*}\left(x,t\right)\left[\frac{d{V}_a}{dx}+E\left(t\right)\right]\psi \left(x,t\right)dx$$

(9)

The corresponding harmonic emission spectrum is obtained by the Fourier transform of the time-dependent dipole moment:

$$P_l\left(\omega \right)={\left|\frac{1}{t_n-t_0}{\int }_{t_0}^{t_n}{X}_{TDSE}\left(t\right)e^{-i\omega t}dt\right|}^2$$

(10)

$$P_a\left(\omega \right)={\left|\frac{1}{{\omega }^2\left(t_n-t_0\right)}{\int }_{t_0}^{t_n}{a}_{TDSE}\left(t\right)e^{-i\omega t}dt\right|}^2$$

(11)

3. Results and Discussion

The code used in the numerical simulation of this article is drafted in FORTRAN language and applied to harmonic optimization. The space size of the simulation box x = 800 a.u., the space step dx = 0.024 a.u., and the time step dt = 0.037 a.u. The accuracy of solving the time-dependent Schrödinger equation using the numerical approach is confirmed by the harmonics in the form of length Equations (8) and (10) and acceleration (Equations (9) and (11)).

Figure 1a depicts the electric field of an ultrashort laser. The wavelength is 800 nm, which has a peak amplitude of 0.08, and the pulse duration of full width at half maximum is 3.5 optical cycles (o.c.). The high-order harmonic generation spectra generated by helium atom are shown in Figure 1b, which are obtained from TDSE with the length (red dotted line) and acceleration (black solid line) forms and calculated from 51 Bohmian trajectories (blue dashed line). The accuracy of solving the time-dependent Schrödinger equation using a numerical approach is confirmed by the harmonics in the form of length and acceleration, as can be seen from the figure. Both close to and far from the core, the wave function is precise. More importantly, the HHG spectra from TDSE can be quantitatively compatible with the harmonic spectra produced by employing 51 Bohmian trajectories. To properly understand the harmonic generation process, a precise trajectory analysis based on quantitative analysis is necessary. The mechanism for generating the different order harmonics is different. Our research can establish the basis for the study of harmonic mechanisms in complex systems. For example, for a polyatomic molecule, the ionized electrons of atoms at different positions in the molecule may recombine with atoms at other positions in the parent ion to generate harmonics, resulting in different harmonic emission mechanisms [48]. The scheme adopted in this article will help to regulate with precision the emission behavior of harmonics. As is shown in Figure 1b, the atomic harmonic spectrum displays typical plateau properties with a cutoff harmonic order of 45. It exhibits different characteristics in different energy ranges: (1) There are distinct odd-order harmonic peaks (41st, 43rd) that can be seen for the harmonics near the cutoff energy, but the associated spectral peaks (1st, 3rd) are broader than the low-energy harmonic peak. (2) The harmonic emission spectrum grows increasingly complicated for the harmonic plateau area. The ultrashort driving laser pulse prevents any discernible odd harmonic emission, which can be observed in the plateau. (3) Even with the ultrashort pulses utilized in this work, obvious odd peaks may be seen since the harmonic efficiency for lower harmonics quickly falls with the order. The transition between the bound states is typically thought to provide a higher harmonic intensity below the ionization threshold. We will then carry out an in-depth analysis of the harmonics close to the cutoff energy, the plateau region, and below the threshold.

Based on the accurate calculation of high-order harmonics, we analyzed in detail the spatial and temporal distribution of the time-dependent dipole moment calculated by the Bohm trajectory using Equation (5). The spatial and temporal distribution of the dipole moment is found that there are larger intensities at both sides of the atom nuclear, as shown in Figure 2. The amplitude of the spatio-temporal distribution oscillates between both sides of the nucleus. For the larger time-dependent laser electric field, the intensity amplitude of the distribution is larger, and the spatial range of the distribution is wider. The behaviors of spatio-temporal distribution can be explained by the evolution of time-dependent wave function. When the atom is irradiated by the laser field, its time-dependent wave function is also oscillating due to the action of the laser electric field [42,51]. With the spatial and temporal distribution of dipole moment, one can obtain the spatial and temporal distribution information of the harmonic emission spectrum through Equation (6).

In Figure 3a, the temporal and spatial distribution of the 45th harmonic is presented. Under the influence of the ultrashort laser pulse, the emission time of harmonics near the cutoff energy occurs mostly after the peak of the driving laser intensity peak. There are two emission peaks in the figure. The width of the peak width is about 0.1 optical cycles, and the instants of the emission can be calculated by the classical three-step model. The electron only ionizes around the electric field’s peak of the driving laser. Then, it returns to the parent ion to try to obtain the most kinetic energy possible, which is combined with the parent ion to produce the emission. As a result, the greater intensity of the harmonic emission is produced when the electron returns to the parent ion. The weighted Bohmian trajectory under the influence of the driving laser is still dispersed close to the nuclear area. Since the Bohmian trajectory behavior is determined by the wave packet, a significant percentage of its contribution to the harmonic computation is admitted. As a result, the nuclear area is where the spatial distribution of the harmonic behavior in the cutoff region is most frequently seen. As a point of reference, we examine the spatial and temporal distribution of the 41st harmonic at the plateau area, as shown in Figure 3b. The long and short emission trajectories can be observed in the figure. The main duration of harmonic emission is ranged from 7.5 to 8.5 optical cycles. In half-cycle, there are two emission trajectories. These two primary emission structures correlate to the emission of long and short trajectories that produce harmonics in a half-cycle, according to the three-step model calculation. The spatio-temporal characterization of harmonic emission becomes more complicated for lower-order harmonics in the plateau area, as is shown in Figure 4a,b. The spatio-temporal behavior of the 35th and 20th harmonics are presented in the figures. For the 35th harmonic, the harmonic emission’s primary region is localized in the range of 7 to 9.5 o.c. One can observe two emission trajectories around 8 o.c. and 8.5 o.c. The spatial range of the distribution of the harmonic is larger than the harmonic, whose energy is the cutoff energy. The greater spatial distribution of harmonic emission is attributed to the role of the bound state driving by the laser pulse. The emission time range of the 20th harmonic emission is further enlarged, as can be seen in Figure 4b. There is clear emission in the range from 7 to 10.5 o.c. It is found that the low-energy partial harmonic is different from the high-energy harmonic in the plateau area. For the lower-energy harmonic, there are no distinct single-cycle emission structures. The emission structure becomes more continuous and complicated. The structure of the lower-energy harmonic can be attributed to the multiple recombinations with the parent ion for the ionized electron. As a result, harmonic emission’s space-time behavior gets more complicated.

The investigation of harmonics below the threshold is also conducted by using the same research strategy as described previously. Figure 5a shows the temporal and spatial distribution of the third harmonic. The variation of its emission behavior is consistent with the change in the driving laser pulse. It is noted that the spatial position of the harmonic emission deviates from the nucleus and is no longer close to the atomic nucleus. The spatial symmetry of the third harmonic emission is obviously different from the harmonics in the platform region. From the spatio-temporal characters of the harmonic, its generated mechanism can be explained as the motion of electrons close to the core driven by a laser pulse. This harmonic emission can be modeled by the electric dipole emission. In Figure 5b, we presented the spatio-temporal emission behavior of the harmonic with high intensity below the threshold. The energy of the harmonic is 0.58 (the energy difference between the ground state and the first excited state). Unlike the behaviors of the harmonic in the plateau area, the main emission duration of this harmonic is behind the action time of the instant of the peak at the driving laser pulse. Its emission duration is concentrated at 7.5–13 o.c. The harmonic’s space-time behavior can well explain by the creation of the resonance excitation transition. Its emission is mostly focused on the falling edge of the pulse. The reason is that the excited state of the system is occupied at the rising edge of the driving laser and then jumps down to the ground state to emit photons.

4. Conclusions

In short, we investigated the behavior of atomic harmonic generation with the precision of angstroms by using the Bohemian trajectory scheme. The Bohmian trajectories are calculated from the wave function, which is obtained from the numerical solution of the time-dependent Schrödinger equation. By using the wavelet of the Bohmian trajectories, the spatial and temporal characteristics of atomic harmonic are calculated. It is found that the low-order harmonic bellowed the ionized threshold is emitted on both sides of the nuclear area and that the harmonic emission in the plateau regime is close to the nuclear area. The spatio-temporal characteristics of the harmonic in the resonance energy show there is a time delay for the harmonic emission. With the scheme proposed in this work, the physical mechanism of the harmonic emission from the complex molecule will be explained.