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Abstract
Accurate emission times of high-order harmonic generation (HHG) are vital for high-precision ultrafast detection in attosecond science, but a quantitative analysis of Coulomb effects on this time is absent in the molecular HHG. Here, we investigate the Coulomb-induced emission-time shift in HHG of $\text {H}_2^+$ with two different internuclear distances R, where the times obtained via the Gabor transform of numerical data from solving the time-dependent Schrödinger equation are used as simulation experiment results. Based on the molecular strong-field approximation, we develop a trajectory-resolved classical model that takes into account the molecular two-center structure. By selecting appropriate electron trajectories and including Coulomb interactions, the classical trajectory method can reproduce Gabor emission times well. This consistence reveals that Coulomb tails cause an emission-time shift of ∼35 as at the R = 2.0 a.u. case and of ∼40-60 as at the R = 2.6 a.u. case under the present laser parameters when compared to the Coulomb-free quantum-orbit model. Our results are of significance to probe the attosecond dynamics via two-center interference.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Observing the motion of microscopic matters on their nature time scale is one of the fundamental tasks of physics. The development of laser technology has made unprecedented progresses on imaging the atomic and molecular worlds by the interaction between matters and lights. When strong laser fields interact with gases, solids, liquids and plasma, many interesting nonlinear responses occur [1,2], such as multi-photon ionization [3], above-threshold ionization [4], nonsequential double-ionization [5], and high-order harmonic generation (HHG) [6–10]. In attosecond physics, high-harmonic spectroscopy (HHS) is a significant branch that aims at detecting ultrafast dynamics of atoms, molecules and solids via HHG [11–17]. For examples, HHS has been applied to study tunneling dynamics [18,19], nuclear vibrational dynamics of molecules [13,20,21], multielectron dynamics [22,23], and the valence electron density in solids [16].
The process of HHG can be well understood by the classical three-step model [24]. First, the electron escapes from the potential barrier depressed by laser fields. The ionized electron is subsequently accelerated by the fields, gaining kinetic energy. Finally, the electron may return to the parent ion, along with the emission of high-frequency photon, i.e., the high-order harmonic radiation. In the second step, the motion of the electron after tunneling is almost classical and thus can be described in terms of trajectories [24,25]. For one electron trajectory, “ionization time" means the moment when the electron tunnels out of potential barriers. “Recombination time" means the moment when the electron recollides with the parent ion. To image the ultrafast evolution process of electrons and nuclei using time-resolved HHS, it is necessary to accurately measure the ionization and recombination times.
In experiments, two commonly used methods for detecting these times via atomic HHS utilize nontrivial forms of laser pulses: (i) One is termed as phase gating, probing recombination (emission) times of the atomic HHG by measuring the strength of even harmonics as a function of the relative phase between the parallel two-color fields [26]. (ii) The other known as orthogonally two-color (OTC) fields scheme, enables one to extract both ionization and recombination times in HHG for helium [27]. Recent theoretical studies show that the improved OTC method is able to reveal the Coulomb shift of ionization times in HHG [28,29]. Molecules, compared to atoms, have additional degrees of freedom and the multicenter nature, leading to complex multichannel HHG [30,31]. Challenge comes with an experimental measurement of both ionization and recombination times using molecular HHS. Therefore, some experiments paid attention to the detection of the molecular harmonic emission time, which is closely related to the attosecond dynamics of nuclear wavepackets [32–34]. For numerical experiments according to solving the time-dependent Schrödinger equation (TDSE), the Gabor transform supplies an effective way to reconstruct the emission time of molecular HHG [35,36].
Apart from the fundamental interest of deeply understanding the concept of electron trajectories in HHG and in high-energy above-threshold ionization [24,37–39], the reconstruction of the harmonic emission time has a wide range of practical implications in strong-field physics. As we know, the interaction of the laser-driven electron wave packet with molecular cores results in quantum interferences [40,41]. Two-center interference, a typical case of such interference in diatomic molecules, leads to a jump of harmonic phases at the harmonic order with minimum yields [42]. Since the harmonic emission time is the derivative of the harmonic phase to the frequency, the phase jump is encoded in the modification of emission times [32]. This indicates that the mapping of the harmonic frequency and the emission time is at the heart of exploring attosecond dynamics through two-center interference [22,32,43–45]. Besides that, a well-defined emission time plays a crucial role in steering an attosecond burst of extreme ultraviolet light [32], monitoring the dynamics of excited states of asymmetry molecules [36] and molecular orbital tomography [46]. The electron-nuclei Coulomb potential, an inherent interaction in HHG, is known to, for example, cause Coulomb-time shift in harmonic radiation [28,39,47,48] and enhance the harmonic emission [49,50]. Nevertheless, most of them are concentrated on atomic and solid systems. For molecular HHG, in which the Coulomb potential is more complex than atoms and simper than solids, a $\mathit {quantitative}$ research on the influence of Coulomb potentials on harmonic emission times has not been reported so far.
In this paper, we study the Coulomb effect on the emission time in HHG from atomic and molecular systems. The emission time obtained by using the Gabor transform of TDSE data is served as numerical experiment results. For a simple hydrogen atom, the Gabor emission time is well reproduced by a Coulomb-modified classical model (CM), which reveals the Coulomb shift of emission times. For the $\mathrm {H}_2^+$ molecules with fixed internuclear distances, a trajectory-resolved CM developed by molecular strong-field approximation (SFA) [51–53] is used to analyze the harmonic emission-time shift induced by Coulomb tails. We find that with a careful selection of the moving path of the electron, the CM results show a good agreement with the TDSE emission time of $\mathrm {H}_2^+$ HHG. These findings imply that the trajectory-based CM is sensitive to the choice of electron trajectories, which is able to quantitatively describe Coulomb-caused emission-time shifts in HHG.
This paper is organized as follows: In Sec. 2, we introduce the detailed procedure of solving the TDSE. In Sec. 3, we develop a Coulomb-corrected classical model based on the strong-field approximation to quantitatively characterize the influence of the Coulomb interaction on emission times of the atomic/molecular HHG. Finally, Sec. 4 is devoted to our conclusions. The interaction between strong laser fields and atoms or molecules is restricted to the tunneling regime.
2. Theoretical method
The interaction of atomic/molecular systems with strong laser fields is described by a two-dimensional (2D) time-dependent Schrödinger equation. In the single-active-electron approximation, the TDSE reads (atomic units are used throughout the paper unless otherwise stated)
with the Hamiltonian in the length gaugeThe driving field $\mathbf {E}(t) = - \partial \mathbf {A}(t)/ \partial t$, polarized along the $x$-axis, is defined by the vector potential
with the unit vector $\mathbf {\hat {e} } _x$, where $E_0$ is the peak amplitude of the electric field. We set the frequency $\omega$ as $0.057$ a.u., corresponding to an $800$-nm laser field. This external field has a trapezoidally envelope $f(t)$ with one-cycle linear rising edge ($-T$ to 0), one-cycle plateau (0 to $T$) and one-cycle linear falling edge ($T$ to $2T$). $T=2\pi /\omega$ donates the optical period of the laser pulse. But for $\mathrm {H}_2^+$, to restrain the HHG from long trajectories, we take the steep form $f(t)=[\mathrm {cos}(\omega t/4)]^{6}$ for the leading ramp and $f(t)=[\mathrm {sin}(\omega t/4)]^{6}$ for the trailing ramp [28,29,54]. We have numerically confirmed that the emission time obtained from the Gabor transform of the TDSE data is independent of the function form of ramps. The present laser pulse guarantees that the time zero point is placed at the beginning of the pulse plateau.The evolution of the TDSE is limited within a box of the size $L_{x} \times L_{y} = 500 \times 200 $ a.u. with $10000 \times 4000$ grid points. The Crank-Nicolson method [55] is used to solve the 2D TDSE with a time step $\Delta t = 0.05 $ a.u. To avoid reflections from the boundary, the wave function $\psi (\mathbf {r},t)$ is multiplied with a mask function $M(x)$ [or $M(y)$]: (i) Along the $x$-axis, we have $M(x) = \mathrm {cos}^{1/8}(\pi (|x| - x_{0} )/(L_{x} - 2 x_{0}))$ for $|x| \geq x_{0}$ and $M(x) = 1$ for $|x| \leq x_{0}$ with $x_{0} = 1.1 E_0/\omega ^{2}$. (ii) In the $y$-direction, $M(y)$ has the same form as $M(x)$ with $y_{0} = 7 L_{y} / 18$. Once obtaining the wave function $\psi (\mathbf {r},t)$, the dipole acceleration $\mathbf {a}(t)$ is calculated via the Ehrenfest theorem [56],
The total high-order harmonic spectrum is calculated from the Fourier transform of the dipole acceleration $a_i(t)\left ( i=x,y \right )$
3. Results and discussion
3.1 Harmonic emission times of the H atom
The high harmonic spectrum of the H atom driven by an 800-nm laser pulse with the intensity $3.51 \times 10^{14} \mathrm {W/cm^{2}}$ is displayed in Fig. 1(a). A cutoff occurs near the expected energy $I_p^a + 3.17U_p\approx 2.94$ a.u. with the ponderomotive energy $U_p=E_0^2/(4\omega ^2)$. Generally, two types of electron trajectories (named as short and long trajectories) contribute to the same-order harmonic generation. Since the harmonic spectrum only contains the frequency domain information, we have to isolate the short trajectory of the harmonic radiation (here we only focus on one short trajectory, because the long trajectory can be suppressed by adjusting the macroscopic phase in experiments) and find the emission time, with the help of the Gabor transform. For this reason, the harmonic spectra are not shown for $\mathrm {H}_2^+$ cases. From the dipole acceleration $a_i(t)$, the total Gabor intensity is the sum of $x$ and $y$ components
Fig. 1. (a) High-harmonic spectra of the hydrogen atom. The laser intensity and wavelength are $3.51 \times 10^{14} \mathrm {W/cm^{2}}$ and $800$ nm, respectively. (b) The Gabor analysis for HHG from the hydrogen atom. The triangles connected by the blue line show the local maxima of the Gabor distribution for the short trajectory in the flattop of laser pulses.
As the first step to analyze the effect of Coulomb tails on emission times, the results calculated from the atomic quantum-orbit model (Atom-QO) [25] without Coulomb potentials, are shown in Fig. 2. In the Atom-QO model, the ionization time $t_i$ and the recombination time $t_r$ are solved by the saddle-point equations
Fig. 2. Harmonic emission times of the H atom model: CM with Coulomb potentials ($\gamma = 1$, black empty squares), CM without Coulomb potentials ($\gamma = 0$, red empty circles), Atom-QO model (green solid lines), TDSE (magenta dashed lines), and SM (blue dashed-dotted lines). The laser parameters are same as Fig. 1.
This shift can be well understood by a classical model. In the classical picture, the electron trajectory is along the direction of the linearly polarized laser field ($x$-axis), obeying the Newton’s equation of motion
here the Coulomb force $-\gamma \, \partial _{\mathbf {x}} V(\mathbf {r})$ is the negative gradient of Coulomb potentials used in the TDSE calculations with $y=0$. The Coulomb factor $\gamma =0/1$ indicates a Coulomb-free/Coulomb-correction case. In both cases, the initial velocity3.2 Harmonic emission times of the $\mathrm {H}_2^+$ molecule
In this section, we study the case of $\mathrm {H}_2^+$ molecules with a frozen internuclear distance $\mathbf {R} =R\,\mathbf {\hat {e} } _x$. The used laser pulse has the wavelength of $800$ nm and the intensity of $3.51 \times 10^{14} \mathrm {W/cm^{2}}$. To quantitatively calibrate Coulomb-induced emission-time shifts in HHG, a benchmark that lacks Coulomb effects is crucial. The Atom-QO model serves as such a reference in the H atom case. Here, the saddle-point equations [Eqs. (8) and (9)], derived from the atomic SFA [58], only lead to one class of possible electron trajectories [see the orange curve in Fig. 3(a). Note that here short and long trajectories belong to the subsets of this electron path]. Unlike the H-atom case, for diatomic molecules, the electron ionized from one potential well and returning to either of the two centers will give rise to four groups of electron trajectories: $\alpha \to \beta$. These trajectories are plotted in Fig. 3(b), labeled as $(\alpha, \beta )$ with $\alpha,\beta \in \left \{ 1,2 \right \}$, where $\alpha$ and $\beta$ correspond to the initial and final atomic center during the electron excursion, respectively. Based on the molecular SFA in Refs. [51–53], in which the multicenter structure of molecules is encoded in the prefactors of the ionization and recombination transition matrix elements, the molecular quantum-orbit model (Mol-QO) has
Fig. 3. Sketch of one-dimensional electron trajectories in HHG of (a) atoms and (b) aligned homonuclear molecules along the laser polarization. The Coulomb potentials $V(x)$ are shown as the gray dashed curves. The red solid curves represent the effective potentials $V(x)+xE(t)$, resulting from a negative laser field $E(t)$. It means the ionization towards the positive $x$-axis. For the H atom, there is only one type of electron trajectories (orange arrowed curves). However, for $\mathrm {H}_2^+$, due to the two-center structure, there are four types of electron trajectories $(\alpha,\beta )$: $(1,1)$, $(1,2)$, $(2,1)$ and $(2,2)$. Physically, $(\alpha,\beta )$ implies that the electron tunnels from the center $\alpha$ and recombines with the center $\beta$.
Before analyzing emission-time shifts caused by double-well potentials, we introduce the trajectory-dependent CM according to the above molecular QO model. Similar to the atomic CM, the motion of an electron in the molecular CM follows the Newton’s equation, i.e. Equation (10). But considering two facts: (1) When using Eq. (12) to evaluate the tunnel exit, the starting position defaults to zero; (2) Two hydrogen atoms in $\mathrm {H}_2^+$ are symmetrically located on both sides of the zero-point along the $x$-axis, the initial position is modified to
Fig. 4. (a) Gabor time-frequency profile of the harmonic generation from $\mathrm {H}_2^+$ with the internuclear distance $R=2.0$ a.u. The local maxima of the Gabor intensity is shown by the blue solid line with triangles. (b) Emission times from various models are shown: Mol-QO with $\alpha =\beta =1$ (green lines), CM with $\gamma =\alpha =\beta =1$ (black empty squares), CM with $\gamma =0$ and $\alpha =\beta =1$ (red empty circles), and the SM model (blue dashed-dotted lines). The magenta dashed line represents the extracted Gabor emission time. The laser intensity of the 800-nm driving field is $3.51 \times 10^{14} \mathrm {W/cm^{2}}$.
Case of the internuclear distance $R=2.0$ a.u. Firstly, we set $R$ as the equilibrium internuclear distance, i.e. $R=2.0$ a.u. The Gabor analysis for HHG from $\mathrm {H}_2^+$ is presented in Fig. 4(a), which isolates a clear-cut short trajectory. The Gabor emission times are plotted in Fig. 4(b). The good agreement of emission times between the TDSE and the CM with $\gamma =1$ reveals that the Coulomb tails result in a shift of about $35$ as earlier when compared to the Mol-QO model. Note that in both theoretical models (Mol-QO and CM) here, the electron is ejected from and recollides with the same center ($\alpha =\beta =1$).
Physically, all four groups of electron trajectories $(\alpha,\beta )$ would contribute to the molecular HHG [51,59]. Nevertheless, Fig. 5 shows: (i) The emission time estimated only by CM $(1,1)$ fully matches those from TDSE. (ii) Remarkably, the CM harmonic orders cannot reach to 60 for trajectories $(2,1)$ and $(2,2)$. The reason is that for higher harmonic orders, the Coulomb force $\left. -\partial _{\mathbf {x}} V(\mathbf {r}) \right |_{\mathbf {x}_\alpha }$ at starting time $t_0$ is larger than the electric field force $- \mathbf {E}(t_0)$ in the CM. This leads to the electron directly moving towards to the molecular center $\beta$ without oscillations in the laser field. It also happens in the $R=2.6$ a.u. case [see the inset in Fig. 7]. This limitation is related to the influence of Coulomb potentials on the tunneling dynamics. Because the Coulomb force $\left. -\partial _{\mathbf {x}} V(\mathbf {r}) \right |_{\mathbf {x}_\alpha }$ acquires larger values for trajectories $(2,\beta )$ due to the tunnel exit $\mathbf {x}_\alpha$ closer to positions of two nuclei when compared to the $\alpha =1$ cases. In other words, the Coulomb effect on tunneling dynamics may become important [60] for trajectories with the electron ionized from the $\alpha =2$ center.
Fig. 5. Harmonic emission times obtained from the CM with different classes of electron trajectories $(\alpha,\beta )$ for $\mathrm {H}_2^+$ with $R=2.0$ a.u., labeled as CM $(\alpha,\beta )$. The magenta dashed line is the Gabor emission time. The inset in the upper left part shows emission times of CM $(2,1)$ and CM $(2,2)$.
Case of the internuclear distance $R=2.6$ a.u. Secondly, a larger internuclear distance $R=2.6$ a.u. is chosen to show the dependence of Coulomb-induced emission-time shifts on $R$. Analogous to the case of $R=2.0$ a.u., the CM $[\gamma =\alpha =\beta =1]$ times match well with the TDSE results as shown in Fig. 6(b). The finding implies a Coulomb-induced emission-time shift of $\sim 40$-$60$ as towards earlier times when using the Coulomb-free times from the Mol-QO model (or the CM with $\gamma =0$) as references. For comparison with the $R=2.0$ a.u. case, the amount of time shifts is larger and is more dependent on the harmonic orders. The larger internuclear distance means the lower ionization potential and the larger energy $\mathbf {E}(t)\cdot \frac {\mathbf {R}}{2}$ in Eqs. (14), (15) and (19). These $R$-dependent factors cause the Coulomb time shift depending on $R$. Fig. 7 also displays the emission times calculated according to the trajectory-resolved CM. These four results [CM (1,1), CM (1,2), CM (2,1) and CM (2,2)] are all different from each other. Obviously, the CM $(1,1)$ shows a good consistence with TDSE results. Hence in this case, there only exists one type of the electron trajectory mainly contributing to HHG, i.e., the pathway where the electron is born at nucleus $\alpha =1$ and recombines at nucleus $\beta =1$.
Fig. 6. Same as Fig. 4, but for the $\mathrm {H}_2^+$ molecule with $R=2.6$ a.u. The laser field has the intensity of $3.51 \times 10^{14} \mathrm {W/cm^{2}}$ and the wavelength of $800$ nm.
Fig. 7. Same as Fig. 5 but for the $\mathrm {H}_2^+$ molecule with $R=2.6$ a.u. All the laser parameters are the same as in Fig. 6.
For completeness, we also present the SM emission times in Fig. 4(b) and in Fig. 6(b). There is a significant difference between SM predictions and results from TDSE, CM and Mol-QO. The zero initial conditions and lacking of Coulomb interactions in the SM are responsible for such difference. In addition, the calculations are carried out at different laser intensities ($3.0 \times 10^{14} \mathrm {W/cm^{2}}$, $4.0 \times 10^{14} \mathrm {W/cm^{2}}$ and $5.0 \times 10^{14} \mathrm {W/cm^{2}}$). The main conclusion remains valid except that the harmonic emission time shift depends on laser intensities. Overall, our results show that the CM is able to quantitatively describe the Coulomb shift via choosing an appropriate electron trajectory $(\alpha,\beta )$.
We don’t discuss the case of larger internuclear distances for $\mathrm {H}_2^+$ molecules in the present work. Because in this case, the excited electronic states and the multichannel interference are possible to play a significant role in molecular harmonic emission processes [61,62], it is a challenge to consider these influences in molecular SFA and to find one single short trajectory using the Gabor analysis of TDSE simulation data. As for $\mathrm {H}_2^+$ with smaller internuclear distances, the semiclassical action in molecular SFA does not feel the molecular structure so that they can be summarized into the atomic case.
4. Conclusions
In summary, we have studied the emission times (recombination times) in HHG from the H atom and the $\mathrm {H}_2^+$ molecules with fixed internuclear distances. Notwithstanding the previous work investigated the influence of Coulomb tails on harmonic emission times from various atomic/molecular systems with short- and long-range potentials, deeper theoretical investigations are necessary [35]. For the simple hydrogen model, the classical electron-trajectory theory can reproduce the Gabor emission time, identifying a time shift of $\sim 20$-$45$ attoseconds earlier than the Coulomb-free QO model, caused by the Coulomb force. However, the electron has the possibility to leave from one center (the ionized center $\alpha$) and recombines with the same center or the other one (the recombined center $\beta$) during $\mathrm {H}_2^+$ HHG from the perspective of classical trajectories, bringing about four types of electron trajectories $(\alpha,\beta )$ with $\alpha,\beta \in \left \{ 1,2 \right \}$. Considering the above fact and using the molecular SFA method in Refs. [51–53], we develop a molecular trajectory-resolved CM and apply it to analyze the Coulomb-induced emission-time shift in two cases of the internuclear distance $R=2.0$ a.u. and $R=2.6$ a.u. for $\mathrm {H}_2^+$. In both cases the TDSE emission times coincide with CM ($\alpha =\beta =1$) for harmonic orders in the plateau. These results determine the electron trajectory that leads to the $\mathrm {H}_2^+$ HHG. Importantly, they reveal that the electron-core Coulomb interaction results in (i) a $\sim 35$-as emission-time shift at the $R=2.0$ a.u. case and (ii) a $\sim$($40$-$60$)-as emission-time shift at the $R=2.6$ a.u. case for $\mathrm {H}_2^+$ molecules.
In experiments, after measuring the emission times of the molecular HHG, the multichannel dynamics in HHG can be resolved on the attosecond time scale with the help of the molecular CM. However, for a real experiment the nuclei in molecules undergo vibrational motion neglected in this paper. But the internuclear distance almost keeps constant if the pulse duration is shorter than the vibrational period. This seems to be possible for any molecule, especially for molecules with heavy nuclei, since the pulse length less than $10$ femtoseconds is available in the laboratory.
Funding
National Natural Science Foundation of China (12204209, 12274188); Natural Science Foundation of Gansu Province (23JRRA1090); Fundamental Research Funds for the Central Universities (lzujbky-2023-ey08).
Acknowledgments
We thank M. Lein, X. Zhu, and B. Zhang for helpful discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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