Enhancement of High-Order Harmonic Generation by Suppressing Quantum Diffusion of the Electron Wavepacket

Abstract

High-order harmonic generation with mid-infrared laser fields has been considered the most promising method to produce soft X-rays attosecond pulses, which provides an important tool for probing the ultrafast electronic dynamics in atoms, molecules, and solids in real time. However, quantum diffusion of the electron wavepacket can lead to a dramatic drop of the harmonic yield when a mid-infrared laser field is used. Here we theoretically demonstrate that a spatially structured (SS) laser field can suppress quantum diffusion of the electron wavepacket and lead to a significant enhancement of high-order harmonic generation, compared with those generated by the spatially homogeneous (SH) laser field. The SS laser field is inhomogeneous in transverse direction perpendicular to the laser polarization and homogeneous in the polarization direction of the laser field. The electric field presents a valley structure. It is found that this valley structure can confine the electron wavepacket around the parent ion, prevent the electron wavepacket spreading, and finally lead to the significant enhancement of the high-order harmonics. Our results provide a novel method for controlling the ultrafast electron wavepacket dynamics of HHG.

1. Introduction

The advent of attosecond pulses has enabled the extraction of dynamic information about electrons in atoms, molecules, and solids in real time, thereby opening new fields of time-resolved metrology on a sub-femtosecond time scale [1,2,3,4,5,6,7]. High-order harmonic generation (HHG) has proven to be a unique method for obtaining extreme ultraviolet or soft X-ray attosecond pulses [8] and has garnered recognition from the scientific community, culminating in a Nobel Prize in Physics awarded in 2023 [9]. The HHG process can be effectively described by the semiclassical three-step model [10]: ionization, acceleration, and recombination of the electrons in an intense laser field. During recombination, high-order harmonic photons are emitted. The maximum photon energy corresponding to the harmonic cutoff adheres to the cutoff law, expressed as $I_p+3.17U_p$, where $I_p$ is the ionization potential of the target, and $U_p\propto I{\lambda }^2$ is the ponderomotive energy of the electron in a laser field of intensity $I$ and wavelength $\lambda$.

Previously, the extension of the harmonic cutoff has attracted considerable attention, as a broader spectrum facilitates the generation of shorter attosecond pulses and enhances the resolution for ultrafast measurements. On one hand, increasing the laser intensity can directly raise ponderomotive energy according to the cutoff law. However, extremely intense driving fields can lead to ionization saturation of the target, thereby limiting the harmonic yield. On the other hand, since the highest generated photon energy is proportional to the square of the driving laser wavelength, many mid-infrared laser schemes have been proposed to extend the harmonic cutoff [11,12,13,14]. Unfortunately, in this context, the harmonic yield is found to decrease dramatically with increasing laser wavelength [15,16,17,18]. It has been shown that a scaling of ${\lambda }^{-3}$ in harmonic yield is due to the spreading of the electron wavepacket [14,15]. In previous studies [19,20,21], effective suppression of electron wavepacket diffusion was demonstrated using vibrationally or electronically excited initial states of atoms and molecules, resulting in an increase in high-order harmonic yield. However, the implementation of such mechanisms necessitates a more complex experimental setup to ensure the preparation of a nonlinear medium in an excited state.

Spatially structured (SS) light, characterized by inhomogeneity in amplitude, phase, or polarization, has attracted widespread attention for its ability to produce nontrivial phenomena and novel functionalities compared to homogeneous light [22,23,24,25,26]. Recently, structured light has been utilized in strong-field physics, particularly in HHG across various scales [27,28,29,30,31,32,33]. SS light presents a potential application in controlling the electron dynamics of high-order harmonics to modulate harmonic spectral properties.

In this paper, we theoretically investigate high-order harmonic generation by SS mid-infrared laser fields. The SS laser field inhomogeneous in transverse direction perpendicular to the laser polarization and homogeneous in the polarization direction of the laser field is considered for the first time. Two kinds of spatial structure are considered here. One possesses a valley structure, while the other possesses a peak structure. It is found that the SS field with a valley structure confines the electron wavepacket around the parent ion and prevents the electron wavepacket spreading. As a result, the harmonic yields are significantly increased compared with those driven by a spatially homogeneous (SH) laser field, whereas the SS field with a peak structure is found to accelerate the diffusion of the electron wavepacket. This dramatically decreases the returning electron trajectories. As a result, the harmonic yields are decreased throughout the harmonic spectrum, compared with those driven by the SH laser field. Our results provide a novel method for controlling the ultrafast electron wavepacket dynamics of HHG.

2. Model and Methods

In this work, we investigate the HHG from atoms in a spatially structured mid-infrared laser field by numerically solving the 2D time-dependent Schrödinger equation (TDSE) (atomic units are used throughout this paper unless otherwise stated):

$$\mathrm{i}{\displaystyle \frac{\partial }{\partial t}}\Psi \left(\overrightarrow{r},t\right) = \widehat{H}(\overrightarrow{r},t)\Psi (\overrightarrow{r},t)$$

(1)

The Hamiltonian reads as follows:

$$\widehat{H}\left(\overrightarrow{r},t\right) = -{\displaystyle \frac{1}{2}}{\mathsf{\nabla }}^2+V(\overrightarrow{r})+\overrightarrow{r}·\overrightarrow{E}(\overrightarrow{r},t) .$$

(2)

$\overrightarrow{E}(\overrightarrow{r},t)$ is the electric field and $V(\overrightarrow{r})$ is the model potential of the target. Note that the space dependence of the laser field cannot be neglected here as we focus on SS fields, while it reduces to a space-independent quantity $\overrightarrow{E}(t)$ for SH fields. Here we use the single-active-electron (SAE) soft-core potential for atom, which is given by the following:

$$\mathrm{V}\left(\overrightarrow{\mathrm{r}}\right)=-{\displaystyle \frac{1}{\sqrt{{\mathrm{r}}^2+{\mathrm{a}}^2}}}$$

(3)

with $\mathrm{a}$ being the soft-core parameter. The TDSE is solved using the split-operator method [34]. The initial state is obtained by the imaginary-time propagation. Then, the time-dependent wavefunction is obtained by real-time propagation, starting from this initial state. According to the Ehrenfest theorem [35], the time-dependent dipole acceleration $\ddot{\overrightarrow{d}}(t)$ is given by the following:

$$\ddot{\overrightarrow{d}}\left(t\right) = -⟨\Psi \left(\overrightarrow{r},t\right)|\mathsf{\nabla }\left(V\left(\overrightarrow{r}\right)+\overrightarrow{r}·\overrightarrow{E}\left(\overrightarrow{r},t\right)\right)|\Psi \left(\overrightarrow{r},t\right)⟩.$$

(4)

After obtaining the dipole acceleration, the harmonic spectrum is given by the Fourier transform of $\ddot{\overrightarrow{d}}(t)$:

$${\overrightarrow{E}}_{XUV} = \int \ddot{\overrightarrow{d}}(t)exp(-iq{\omega }_{0}t)dt,$$

(5)

$$S_q={\left|{\overrightarrow{E}}_{XUV}\right|}^2 ,$$

(6)

where $q$ corresponds to the harmonic order.

3. Results

In this section, we will investigate the effect of the SS character of the laser field on HHG and the corresponding electron dynamics. Our coordinate system defines the polarization vector of the driving pulse as the $x$ axis; the driving pulse propagates along the $z$ axis. The SS laser field is inhomogeneous along the $y$ axis and homogeneous along the polarization direction of the laser field. Hence, the spatial dependent electric field of the driving pulse is given by the following:

$$\overrightarrow{E}\left(\overrightarrow{r},t\right) = \epsilon (y)E(t)\overrightarrow{i}.$$

(7)

In Equation (7), $\epsilon (y)$ indicates the space dependence of the laser field. The development of high-quality metasurface nanostructures offers an approach to generate a SS field [22,23,24,25,26]. The spatial distribution of the field amplitude can be modulated by designing the structure of the metasurface. Here, two kinds of spatial structure are considered and are shown in Figure 1 by the blue and the dotted black curves, respectively. For comparison, the transverse structure of the SH field with $\epsilon (y)$ = 1 is demonstrated in Figure 1 by the dashed red curve. From Figure 1, one can see that there is a valley around the core for the SS field 1 and a peak for the SS field 2. The amplitude of the SS fields and the SH field have nearly the same value around the core from −20 a.u. to 20 a.u. The transverse distribution of the ground state wavefunction is also depicted in Figure 1 by the green curve. One can see that the distribution of electron wavefunction is in the range of −20 a.u. to 20 a.u. This indicates that the main part of the electron wavepacket experiences the same laser field in the three cases of the SS fields and the SH field, except for the part that spreads out due to the quantum diffusion effect. Hence, the electron dynamics process (i.e., ionization, acceleration, and recombination) of HHG is identical for the three lasers. By comparison with the case of the SH field, we can investigate the influence of the SS fields on the diffusion part of the electron wavepacket and finally on the harmonic spectrum.

The time-dependent part of the laser field in Equation (7) is expressed as follows:

$$E\left(t\right) ={ E}_{0}f(t)sin({\omega }_{0}t)$$

(8)

where $E_0$ is the amplitude of the laser field, $f(t)$ is the envelope of the laser pulse, and ${\omega }_0=0.015\mathrm{a}.\mathrm{u}.$ is the frequency corresponding to the wavelength of 3 μm. A 6-cycle trapezoidal pulse with a 2-cycle linear rise and a 2-cycle linear fall is used here. The intensity corresponding to the $E_0$ amplitude of the laser field is $1\times {10}^{14}\mathrm{W}{\mathrm{c}\mathrm{m}}^{-2}$. It is important to note that this intensity refers specifically to the local field intensity, rather than the incident laser intensity, which may be several orders of magnitude smaller and allow the nanoplasmonic targets to withstand thermal damage.

We take the argon atom as an example and the soft-core parameter $\mathrm{a}$ takes the value of 0.63 to simulate the ionization potential of the argon atom (15.76 eV). In Figure 2, the harmonic spectra driven by the SS fields and the SH field are presented. From Figure 2, two obvious characteristics are observed. Firstly, the harmonic spectra have a very broadband plateau, and the cutoffs of all the three spectra are located at the 681st harmonics (corresponding photon energy is about 281 eV). The same cutoffs of the three harmonic spectra indicate an identical electron dynamics process of HHG, which is consistent with the discussion of Figure 1. The second characteristic shown in Figure 2 is an obvious enhancement of the harmonics driven by the SS field 1 throughout the spectrum, compared with the harmonic spectrum driven by the SH field. From 600th order to 670th order, the harmonic intensities are enhanced by an order of magnitude, as shown by the blue double arrow. In the case of the SS field 2 shown by the dashed black curve, a suppression of the harmonic intensity is observed throughout the spectrum.

To obtain clear insight into the harmonic enhancement and suppression of the SS fields, we further investigate the emission times of harmonics according to the time-frequency analysis for the three spectra in Figure 2. The results for the SS field 1, the SH field, and SS field 2 are presented in Figure 3a–c, respectively. The same color bar is used in Figure 3a–c. It is obvious that the color in Figure 3a is much brighter than that in Figure 3b, especially for the harmonics ranging from 600th order to 670th order. This implies that the number of quantum trajectories contributing to the harmonic emission for the SS field 1 is much larger than that for the SH field. Hence, a much more intense harmonic radiation is obtained in the case of the SS field 1, as shown in Figure 2. For the 600th order to 670th order harmonics, there are four dominating emission peaks. By comparison, one can find that the number of long quantum trajectories (the right branch of emission peaks) contributing to the harmonic emission decreases significantly from Figure 3a–c. In the case of the SS field 2, the long quantum trajectories even disappear, and only short quantum trajectories contribute to the harmonic emission.

To find out the underlying mechanism of the variation in quantum trajectories contributing to the harmonic emission induced by the SS fields, we analyze the time evolution of the electron wavepacket driven by the SS fields and the SH field. The distribution of the laser field along the polarization direction is the same for the three laser pulses as shown in Figure 1, thus the electron wavepacket dynamics along the polarization direction are the same, which results in the same cutoff of the spectra shown in Figure 2. Hence, the time-dependent transverse distribution of the electron wavepacket at $x =0$ is analyzed. The results are shown in Figure 4a–c for the SS field 1, the SH field, and the SS field 2, respectively. The same color bar is used and represents the square modulus of the electron wavefunction in logarithmic scale. As shown in Figure 4a,b, the electron wavepacket is spread from −100 a.u. to 100 a.u. in the case of the SH field but is spread in the range of −50 a.u. to 50 a.u. in the case of the SS field 1. Comparing the structure of the SS field 1 and the SH field, one can find that the valley structure of the SS field 1 acts as a trap of the electron wavepacket and prevents the electron wavepacket spreading. Hence, the electron wavepacket is confined around the parent ion in the range of −50 a.u. to 50 a.u. As a result, more quantum trajectories contribute to the harmonic emission in the case of the SS field 1. Correspondingly, the harmonic intensity is significantly enhanced compared with that driven by the SH field. In the case of the SS field 2 shown in Figure 4c, the electron wavepacket is widely spread from −200 a.u. to 200 a.u. This is because the peak structure of the SS field 2 accelerates the quantum diffusion of electron wavepacket, leading to a wide spread of the electron wavepacket in transverse direction. Hence, much less quantum trajectories contribute to the harmonic emission, and the corresponding harmonic intensity is suppressed, compared with the case of the SH field. It is demonstrated that the electron wavepacket spreads as ${\tau }^{3/2}$, where $\tau$ is the time the electron wavepacket spends in continuum state [15]. Hence, the long quantum trajectory with a large $\tau$ presents more serious quantum diffusion than the short quantum trajectory. As a result, the contributions of the long quantum trajectory to the harmonic emission are more sensitive to the structure of the SS field. Hence the contribution of the long quantum trajectory decreases significantly from SS field 1 to the SS field 2 where the long quantum trajectory disappears.

To better characterize the effect of the SS field 1 on the electron wavepacket, the two-dimension (2D) electron wavefunction at the interval from $t = 2.778T_0$ to $t = 3.101T_0$ is depicted in Figure 5 and Figure 6 for the SS field 1 and the SH field, respectively. The interval corresponds to the second dominating emission peak contributing to the harmonics from 600^th order to 670^th order, as indicated by the dashed red lines in Figure 3. As shown in Figure 5, the electron wavepacket driven by the SS field 1 is confined near to the parent ion and quantum diffusion is significantly suppressed. In the case of the SH field demonstrated in Figure 6, quantum diffusion of the electron wavepacket is clearly observed and leads to the decreasing number of the quantum trajectories contributing to the harmonic emission.

To consider a realistic case of HHG from an ensemble of atoms, the average effect of the atomic ensemble should be taken into account. As shown in Figure 1, the valley structure of the laser field covers an area of one-third of the total area. Hence, in a uniform distribution of the target atom, the high-order harmonics generated from about one-third of the ensemble of atoms are enhanced approximately by an order of magnitude for most of the harmonics, as shown in Figure 2. Hence, the enhancement of the harmonics due to the suppression of electron wavepacket diffusion still exists after considering the average effect of the ensemble.

4. Conclusions

In this paper, we theoretically investigate high-order harmonic generation with the SS mid-infrared laser fields by solving the 2D time-dependent Schrödinger equation (TDSE). The SS laser field is inhomogeneous in transverse direction perpendicular to the laser polarization and is homogeneous in the polarization direction of the laser field. Two kinds of spatial structure are considered. One presents a valley structure, while the other possesses a peak structure. It is found that the SS field with a valley structure confines the electron wavepacket around the parent ion and prevents the electron wavepacket from spreading. As a result, the harmonic intensities are significantly enhanced compared with those driven by the SH laser field, whereas the SS field with a peak structure is found to accelerate quantum diffusion of the electron wavepacket leading to the suppression of the harmonic yields. Because these phenomena are induced by spatial structures of laser fields, they are robust when the peak intensity of laser varies. Our results provide a novel method for controlling the ultrafast electron wavepacket dynamics of HHG.