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阿秒钟技术为研究原子分子强场电离过程中电子超快动力学提供了强有力的工具. 该技术依赖定量理论模型从实验测量得到的光电子频谱反演电离过程中系统的超快时域信息, 而构建定量强场理论模型的关键问题之一是库仑效应的理论描述. 与原子相比, 分子具有多中心库仑势, 在外场中具有许多新的效应, 这为检验定量理论模型的适用性提供了更广阔平台. 本文针对拉伸到大核间距的分子的阿秒钟特征观测量, 比较了不同理论模型的预言, 指出最近提出的半经典响应时间理论预言与数值实验结果更一致. 该理论强调, 与具有经典性质的远核库仑效应相比, 具有量子性质的近核库仑效应对分子隧穿电离影响更显著. 本文进一步展望了该理论模型在具有电荷共振、固有偶极、取向效应以及复杂库仑势形式的分子强场电离中的应用.
The attosecond technology provides a powerful tool for studying the ultrafast dynamics of electrons during the strong-field ionization of atoms and molecules. This technology relies on quantitative theoretical models to invert the ultrafast time-domain information of the system in the ionization process from the photoelectron spectra obtained through experimental measurements. One of the key issues in constructing quantitative strong-field theoretical models is the theoretical description of the Coulomb effect. The Coulomb potential of molecules, compared with the single-center Coulomb potential of atoms, exhibits a multi-center distribution. This fundamental geometric structure feature results in many new effects of molecules in the external field, such as orientation effect, charge resonance effect, intrinsic dipole effect, and vibration effect. Therefore, it can be expected that the tunneling ionization process of molecules contains more phenomena than that of atoms, which is worthy of in-depth study in experiment and theory. Especially for stretched molecular ions, such as $ {\mathrm{H}}_{2}^{+} $, those exhibiting charge resonance effects in external fields, the difference between near-nucleus and far-nucleus Coulomb effects, which is of great significance for constructing quantitative theoretical models, becomes more complex, providing a platform for testing the applicability of quantitative theoretical models. This work systematically compares the predictions of different theoretical models for the attoclock characteristic observables in molecular systems with large internuclear distances. Through comparative analysis, it is found that the recently proposed semiclassical response time theory, which incorporates near-nucleus Coulomb corrections, shows better agreement with numerical experimental results than the developed strong-field approximation models that consider far-nucleus Coulomb corrections. The semiclassical response time theory establishes a theoretical framework for describing strong-field ultrafast ionization dynamics of stretched molecular systems by considering dual-center Coulomb potential corrections and excited-state contributions. Specifically, it approximates the complex four-body interactions (electron-laser-dual nuclei) in stretched molecular systems to a three-body interaction (electron-laser-dressed-up barrier-proximal nucleus), while using the influence of the other nucleus on the potential barrier as a correction term for the tunnel-exit position. This framework highlights the significant influence of quantum-property-dominated near-nucleus Coulomb effects on molecular tunneling ionization. Furthermore, the theory provides an explicit formula for the response time determined by fundamental laser and molecular parameters. By calculating this response time, the values of attoclock observables are deduced from the theory, thus enabling a clear discussion of ionization time delays in stretched molecular tunneling ionization and revealing that such delays reflect the timescale of strong four-body interactions between the laser, electron, and molecular nucleus. In contrast, the developed strong-field approximation model that simultaneously considers excited-state effects and numerically solves Newton’s equations to describe far-nucleus Coulomb effects cannot fully describe the above-mentioned four-body interaction, making it difficult to quantitatively describe the complex tunneling ionization dynamics under the combined action of coulomb and excited states. Additionally, since this model cannot clearly define the ionization time, the related ionization time delay issues cannot be well discussed. Computational results show that the semi-classical response time theoretical model has improved in terms of calculation accuracy and efficiency, thereby verifying the applicability of this theoretical model in the study of molecular ultrafast ionization dynamics. Moreover, for $ {\mathrm{H}}_{2}^{+} $ with intermediate internuclear distances, the charge resonance effect induces a significant ionization enhancement effect. We present relevant numerical experimental attoclock results and explore the potential applications of the response time theory in such systems. We also envision the extension of this theory to strong-field tunneling ionization in polar molecules, multi-center linear molecules, planar and three-dimensional molecules, and oriented molecules, where interference and Coulomb-acceleration effects compete with each other. -
图 1 不同激光和分子参数下的势函数曲线[40] (a) 在$ y=0 $处$ R=2 $ a.u.的$ {\mathrm{H}}_{2}^{+} $(红线)和模型原子(黑线)的激光缀饰势$ V'\left(\boldsymbol{r}\right)=V\left(\boldsymbol{r}\right)-{E}_{0}x $(实线)和无激光势$ V\left(\boldsymbol{r}\right) $(虚线). 水平线表示$ -{I}_{p}=-1.11 $ a.u.的能量. 插图显示了分子和原子出口位置之间差异结果的放大图. (b) 在$ y=0 $处$ R=16 $ a.u.的$ {\mathrm{H}}_{2}^{+} $的激光缀饰势$ V'\left(\boldsymbol{r}\right)=V\left(\boldsymbol{r}\right)-{E}_{0}x $(实线)和无激光势$ V\left(\boldsymbol{r}\right) $(虚线). 两条水平线分别表示$ {E}_{1}'\approx -\left[{I}_{\mathrm{p}}-{E}_{0}\left(R/2\right)\right]=-0.09 $ a.u.和$ {E}_{2}'\approx -\left[{I}_{\mathrm{p}}+{E}_{0}\left(R/2\right)\right]=-2.13 $ a.u.的能量. 坐标$ {x}_{2}\left({x}_{4}\right) $对应于电子从左(右)核附近向上(向下)缀饰的势垒中隧穿的出口位置. 坐标$ {x}_{1} $和$ {x}_{3} $表示分子左边和右边原子核的位置. 水平的粉红色箭头表示隧穿的方向, $ {{r}_{0}''}_{}=\left|{x}_{2}-{x}_{1}\right|({r}_{0}''=\left|{x}_{4}-{x}_{3}\right|) $表示出口位置相对于左(右)核位置的绝对值. 这里使用的激光振幅为$ {E}_{0}=0.13 $ a.u.
Fig. 1. Potential function curves obtained at different laser and molecular parameters [40]. (a) Sketch of the laser-dressed potential $ V'\left(\boldsymbol{r}\right)=V\left(\boldsymbol{r}\right)-{E}_{0}x $ (solid curves) and laser-free potential $ V\left(\boldsymbol{r}\right) $ (dash-dotted line) for $ {\mathrm{H}}_{2}^{+} $ with $ R=2 $ a.u. (red curves) and the model atom (black curves) at $ y=0 $. The horizontal line indicates the energy of $ -{I}_{\mathrm{p}}=-1.11 $ a.u. The inset shows a close-up of the results for the difference in exit position between the molecule and the atom. (b) Sketch of the laser-dressed potential $ V'\left(\boldsymbol{r}\right)=V\left(\boldsymbol{r}\right)-{E}_{0}x $ (solid curves) and laser-free potential $ V\left(\boldsymbol{r}\right) $ (dash-dotted lines) $ {\mathrm{H}}_{2}^{+} $ with $ R=16 $ a.u. at $ y=0 $. Two horizontal lines represent the energy $ {E}_{1}'\approx -\left[{I}_{\mathrm{p}}-{E}_{0}\left(R/2\right)\right]=-0.09 $ a.u. and the energy $ {E}_{2}'\approx -\left[{I}_{\mathrm{p}}+{E}_{0}\left(R/2\right)\right]=-2.13 $ a.u., respectively. The coordinate $ {x}_{2}\left({x}_{4}\right) $ corresponds to the exit position of electrons tunneling out of the dressed-up (down) potential barrier neighboring the left (right) nucleus. The coordinates $ {x}_{1} $ and $ {x}_{3} $ represent the positions of the left and right nuclei of the molecule. The horizontal pink arrows indicate the direction of tunneling and $ {r}_{0}''=\left|{x}_{2}-{x}_{1}\right|({r}_{0}''=\left|{x}_{4}-{x}_{3}\right|) $ represents the absolute value of the exit position relative to the position of the left (right) nucleus. The laser amplitude used here is $ {E}_{0}=0.13 $ a.u..
图 2 不同方法得到(a)—(d)模型原子、(e)—(h) $ R=2 $ a.u.的$ {\mathrm{H}}_{2}^{+} $、以及(i)—(l) $ R=16 $ a.u.的$ {\mathrm{H}}_{2}^{+} $的光电子动量分布[40] (a), (e), (i) TDSE; (b), (f) SFA; (j) DSFA; (c) A-TRCM方法; (g) S-TRCM方法; (k) L-TRCM方法; (d), (h) MSFA; (l) U'-MDSFA. 值得注意的是, (l)中的U'-MDSFA模型规定向上缀饰势垒附近原子核的位置为隧穿位置矢量的起点(该矢量终点为隧穿出口位置), 并将电子隧穿后演化的初始位置修正为$ \left|\boldsymbol{r}'\left({t}_{0}\right)\right|\equiv {r}_{0}'=\left|{\boldsymbol{r}}_{0}\right|-{r}_{\mathrm{d}} $, 其中$ {r}_{\mathrm{d}}={r}_{\mathrm{p}}-{r}_{0}'' $, $ {r}_{0}''=\left|{x}_{2}-{x}_{1}\right| $. PMD的非零偏移角$ \theta $也在每个面板中表示. 激光参数为$ I=1\times {10}^{15} $ W/cm2, $ \lambda =800 $ nm, $ \varepsilon =0.87 $
Fig. 2. PMDs of (a)–(d) the model atom, (e)–(h) $ {\mathrm{H}}_{2}^{+} $ with $ R=2 $ a.u., and (i)–(l) $ {\mathrm{H}}_{2}^{+} $ with $ R=16 $ a.u., obtained with different methods[40]: (a), (e), (i) The TDSE; (b), (f) the SFA; (j) the DSFA; (c) the TRCM method; (g) the S-TRCM method; (k) the L-TRCM method; (d), (h) the MSFA; (l) the U'-MDSFA. It is noteworthy that the U'-MDSFA simulation in (l) assumes taking the position of the atomic nucleus near the dressed-up potential barrier as the starting point of the tunneling position vector (the endpoint of this vector is the position of the tunnel exit), and modifies the initial position for post-tunneling electron evolution to $ \left|\boldsymbol{r}'\left({t}_{0}\right)\right|\equiv {r}_{0}'=\left|{\mathbf{r}}_{0}\right|-{r}_{\mathrm{d}} $, with $ {r}_{\mathrm{d}}={r}_{\mathrm{p}}-{r}_{0}'' $ and $ {r}_{0}''=\left|{x}_{2}-{x}_{1}\right| $. The nonzero offset angle $ \theta $ of the PMD is also indicated in each panel. The laser parameters are $ I=1\times {10}^{15} $ W/cm2, $ \lambda =800 $ nm and $ \varepsilon =0.87 $.
图 3 MDSFA以及不同修正MDSFA模型预测$ R=16 $ a.u.的$ {\mathrm{H}}_{2}^{+} $的光电子动量分布 (a) MDSFA模型, 规定以坐标原点为隧穿位置矢量的起点(该矢量终点为隧穿出口位置); (b), (c) U(D)-MDSFA模型, 规定以向上(向下)缀饰势垒附近原子核所在的位置为隧穿位置矢量的起点; (d) D'-MDSFA模型, 该模型在D-MDSFA基础上将电子隧穿后演化的初始位置修正为$ \left|\boldsymbol{r}'\left({t}_{0}\right)\right|\equiv {r}_{0}'=\left|{\boldsymbol{r}}_{0}\right|-{r}_{\mathrm{d}} $, 其中$ {r}_{\mathrm{d}}={r}_{\mathrm{p}}-{r}_{0}'' $, $ {r}_{0}''=\left|{x}_{4}-{x}_{3}\right| $. PMD的非零偏移角$ \theta $也展示在每个面板中. 激光参数为$ I= $$ 1\times {10}^{15} $ W/cm2, $ \lambda =800 $nm, $ \varepsilon =0.87 $
Fig. 3. The PMDs of $ {\mathrm{H}}_{2}^{+} $ with $ R=16 $ a.u. calculated by MDSFA and different modified MDSFA models. (a) MDSFA model stipulates that the origin of the coordinate system is taken as the starting point of the tunneling position vector (the endpoint of this vector is the position of the tunnel exit). (b), (c) U(D)-MDSFA model stipulates that the position of the atomic nucleus near the dressed-up (dressed-down) barrier is defined as the starting point of the tunneling position vector. (d) D'-MDSFA model which modifies the initial position for post-tunneling electron evolution to $ \left|\boldsymbol{r}'\left({t}_{0}\right)\right|\equiv {r}_{0}'=\left|{\boldsymbol{r}}_{0}\right|-{r}_{\mathrm{d}} $, with $ {r}_{\mathrm{d}}={r}_{\mathrm{p}}-{r}_{0}'' $ and $ {r}_{0}''=\left|{x}_{4}-{x}_{3}\right| $, based on the D-MDSFA model. The nonzero offset angle $ \theta $ of the PMD is also indicated in each panel. The laser parameters are $ I=1\times {10}^{15} $ W/cm2, $ \lambda =800 $ nm and $ \varepsilon =0.87 $.
图 4 TDSE、L-TRCM模型、U-MDSFA模型、U'-MDSFA模型、D-MDSFA模型和D'-MDSFA模型对于不同R预测的$ {\mathrm{H}}_{2}^{+} $的偏移角. 激光参数为$ I=1\times {10}^{15} $ W/cm2, $ \lambda =800 $nm, $ \varepsilon =0.87 $
Fig. 4. Offset angles of $ {\mathrm{H}}_{2}^{+} $ predicted by the TDSE, the L-TRCM model, the U-MDSFA model, the U'-MDSFA model, the D-MDSFA model, and the D'-MDSFA model for different R. The laser parameters are $ I=1\times {10}^{15} $W/cm2, $ \lambda =800 $ nm and $ \varepsilon =0.87 $.
图 5 对中间核间距$ {\mathrm{H}}_{2}^{+} $隧穿电离的研究 (a) 在$ y=0 $处$ R=4 $ a.u.(绿线)和$ R=6 $ a.u.(红线)的$ {\mathrm{H}}_{2}^{+} $的激光缀饰势$ V'\left(\boldsymbol{r}\right)=V\left(\boldsymbol{r}\right)-{E}_{0}x $(实线)和无激光势$ V\left(\boldsymbol{r}\right) $(虚线). 水平线表示$ -{I}_{\mathrm{p}}=-1.11 $ a.u.的能量. (b), (c)分别是通过TDSE得到的$ R=4 $ a.u.和$ R=6 $ a.u.的$ {\mathrm{H}}_{2}^{+} $的光电子动量分布. PMD的非零偏移角$ \theta $也在每个面板中表示. 电离能均为$ {I}_{\mathrm{p}}=1.11 $ a.u.. 为满足不同系统对电离率的要求, (b)中电场强度为$ I=4\times {10}^{14} $ W/cm2, (c)中电场强度为$ I=3\times {10}^{14} $ W/cm2. 其他激光参数为$ \lambda =800 $ nm, $ \varepsilon =0.87 $
Fig. 5. Research on tunneling ionization of $ {\mathrm{H}}_{2}^{+} $ with intermediate $ R $. (a) Sketch of the laser-dressed potential $ V'\left(\boldsymbol{r}\right)= $$ V\left(\boldsymbol{r}\right)-{E}_{0}x $(solid curves) and laser-free potential $ V\left(\boldsymbol{r}\right) $ (dash-dotted line) for $ {\mathrm{H}}_{2}^{+} $ with $ R=4 $ a.u. at $ y=0 $. The horizontal line indicates the energy of $ -{I}_{\mathrm{p}}=-1.11 $ a.u.. PMDs of $ {\mathrm{H}}_{2}^{+} $ with (b) $ R=4 $ a.u. and (c) $ R=6 $ a.u.. The offset angle $ \theta $ of the PMD is also indicated in each panel. The ionization energy are both $ {I}_{\mathrm{p}}=1.11 $ a.u.. To satisfy the ionization rate requirements of different systems, the electric field strength used in (b) is $ I=4\times {10}^{14} $ W/cm2 and in (c) is $ I=3\times {10}^{14} $ W/cm2. Other laser parameters are $ \lambda =800 $ nm, $ \varepsilon =0.87 $.
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