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The response time of the electron to light in photoemission is difficult to define and measure. The tunneling ionization of atoms and molecules in a strong laser field is a type of strong field-induced photoelectric effect. In this process, the electron response time will change the time of high-order harmonic generation (HHG), which will have a fundamental influence on the reconstruction of electron attosecond dynamics through HHG. We propose a simple theory to resolve the response time problem in strong field atomic tunneling ionization. The response time corresponds to the strong interaction time of three bodies i.e. Coulomb, electron and laser field, which can be determined at the quantum-classical boundary. The observable directly obtained through response time can quantitatively reproduce a series of attoclock experimental curves and provide consistent explanations for these experimental phenomena. This work introduces the main conclusions of response time theory and summarizes in detail the research progress of this theory. Firstly, this theory can be applied to the orthogonal two-color laser field to quantitatively explain the main characteristic structures of photoelectron momentum distribution (PMD). Besides, with this response time theory, the scaling law of the observable in attoclock experiment can be obtained. The proposal of scaling law is expected to provide a systematical theoretical guide for better understanding the applicability or feasibility of the attoclock under different conditions. In addition, based on the atomic response time theory, we further consider the property of multi-center Coulomb potential of molecular and develop a response time theory suitable for molecular system. Subsequently, we further apply the response time theory to polar molecules, by utilizing the asymmetry of PMD closely related to response time to recognize the permanent dipole (PD) effect within the laser sub-cycle. In the end, we discuss the prospects for research on response time. Firstly, it is envisioned to further apply response time theory to weak light and single photon transition to detect the response time of related processes. Besides, considering the significant influence of response time on the property of time-domain of HHG electron trajectories, the recombination (re-scattering) effect based on the current strong field tunneling ionization response time theory can be further investigated, thus extending this theory to describing HHG and above threshold ionization (ATI) processes. Furthermore, designing the “re-scattering electron trajectories” reconstruction scheme based on the electron trajectories with response time correction will provide important suggestions for HHG spectroscopic experiments. Finally, considering the asymmetric ionization caused by the PD effect of polar molecules, if the net ionization yield of adjacent sub-cycles is used as the current indicator, polar molecules can be used as a “micro diode” to study a type of attosecond response switching device. Polar molecular diodes emit electrons through tunneling ionization in laser field. According to the response time theory, tunneling occurs almost instantaneously, and response time needs considering only at the tunneling exit. Based on this, by searching for suitable materials (such as two-dimensional materials), it is possible to design a type of semi-classical diode (which can utilize tunneling) with femtosecond or even sub-femtosecond response time. The response time theory can provide a convenient theoretical tool for designing of such tunneling diodes.
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Keywords:
- photoelectric effect /
- tunneling ionization /
- response time /
- attosecond measurement
[1] Keldysh L V 1965 Sov. Phys. JETP 20 1307
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图 1 强场隧穿电离过程示意图及光电子动量分布结果比较[43] (a) 强场隧穿电离示意图, 电子在
$ {t_0} $ 时刻隧穿出垒, 对应隧穿过程(I); 响应过程(II)发生在$ {t_0} $ —$ {t_{\text{i}}} $ 之间;$ {t_{\text{i}}} $ 时刻之后, 电子电离并做经典运动, 对应过程(III). (b) TDSE光电子动量分布, 激光参数以及偏转角大小如图所示; (c) SFA光电子动量分布; (d) TRCM光电子动量分布, 偏转角θ = 7.7°. 图(c), (d)的激光参数与图(b) TDSE一致, 且图(d)中TRCM参数选择$ {n_{\text{f}}} = 2 $ Fig. 1. Sketch of strong field tunneling ionization process and comparison of PMD results[43]. (a) Sketch of strong field tunneling ionization, electron exits the barrier at the peak time t0 of the laser field, in (I) process, the response process (II) occurs between the exit time
$ {t_0} $ and the ionization time$ {t_{\text{i}}} $ ; after time$ {t_{\text{i}}} $ , the electron makes a classical motion, corresponding to the process (III). (b) PMD result of TDSE, laser parameters and offset angle are as shown; (c) PMD result of SFA; (d) PMD result of TRCM and the offset angle is also shown. Panels (c) and (d) use the same laser parameters as panel (b) TDSE, and the parameter$ {n_{\text{f}}} = 2 $ is selected in TRCM.
图 2 响应时间理论应用于H (a), (b)和He (c) [43]. 图中的实、虚线是响应时间理论预言的结果, 点和圈是实验或TDSE的结果. 需要强调, 图中的Eq. (1)— Eq. (3)依次对应于本文(6)式、(8)式、(9)式. (a) 响应时间理论计算所得偏转角与文献[26]实验(Exp. H)和TDSE结果(Coulomb 1和Coulomb 2)的比较. (b) 响应时间理论计算所得延迟时间与文献[26]实验(Exp. H)和TDSE结果(Coulomb 1和Coulomb 2)的比较. (c) 响应时间理论计算得到的偏转角(包含绝热版本)与文献[28]中的实验(Exp.)以及三维TDSE结果的比较
Fig. 2. Application of response time theory to H (a), (b) and He (c) [43]. Real and dashed lines are the predicted results of response time theory, while the points and circles are the results of experiments or TDSE. It should be emphasized that Eq. (1)–Eq. (3) in these figures, corresponding to the Eq. (6), Eq. (8), Eq. (9) introduced in the theoretical section: (a) Comparison of the offset angles calculated by response time theory with the experimental (Exp. H) and TDSE (Coulomb 1 and Coulomb 2) results in Ref. [26]; (b) comparison of lag time calculated by response time theory with experimental (Exp. H) and TDSE (Coulomb 1 and Coulomb 2) results in Ref. [26]; (c) comparison of the offset angles calculated by response time theory with the experiment (Exp.) in Ref. [28] and our three-dimensional TDSE results.
图 3 响应时间理论应用于其他气体靶[43]. 图中的灰实线和橙色点线是TRCM理论给出的偏转角结果; 蓝色点为实验以及TDSE的结果; Eq. (1)对应于理论部分介绍的(6)式 (a) 与实验He结果(Exp. He) [29]比较; (b) 与实验Ar结果(Exp. Ar) [29]比较; (c) 与实验H2结果(Exp. H2) [30]比较; (d) 与H的TDSE结果(TDSE H) [27]比较. 在所有情况中, TRCM理论均很好地预言了实验的偏转角
Fig. 3. Application of response time theory to other gas targets[43]. Gray solid lines and orange dotted lines are the offset angle results of TRCM theory, while the blue dots represent the results of experimental and TDSE; Eq. (1) in these figures, corresponding to the Eq. (6) introduced in the theoretical section: (a) Comparison between theoretical angles with the He experimental results (Exp. He) [29]; (b) comparison between theoretical angles with the Ar experimental results (Exp. Ar) [29]; (c) comparison between theoretical angles with the H2 experimental results (Exp. H2) [30]; (d) comparison between theoretical angles with the TDSE results (TDSE H) of H [27]. In all cases, the TRCM theory predicts the offset angle of the experiment well.
图 4 应用响应时间理论得到阿秒钟椭偏依赖的标度律关系[46]. 实线是TRCM理论模型的结果, 空心符号分别是TDSE和实验[47]的结果 (a), (b) 动量分布最亮点(最可几轨道)对应沿着激光偏振主轴方向
$ {p}_{x} $ 和副轴方向$ {p}_{y} $ 的动量比较; (c) 动量分布偏转角结果的比较Fig. 4. Applying response time theory to obtain the scaling law for the ellipticity dependence of the observable in attoclocks[46]. The solid line is the result of the TRCM, while the hollow symbol are the results of TDSE and experimental[47], respectively: (a), (b) Comparison of the momentum along the laser polarization main axis
$ {p}_{x} $ and the minor axis$ {p}_{y} $ corresponding to the brightest point (most probable route) of the PMD; (c) comparison of the results of PMD offset angles.
图 5 响应时间理论应用于正交双色激光场[48] (a) TDSE动量分布结果; (b) 响应时间理论(TRCM)结果; (c) 库仑修正的强场近似(MSFA)结果; (d) 强场近似(SFA)结果. 图中水平白线展示TRCM预言肩结构对应位置数值, 垂直双白线中间部分是矩形结构, 水平红色箭头指示矩形结构的宽度. 可以看到TRCM肩结构及矩形结构的位置与TDSE的结果定量上一致
Fig. 5. Application of response time theory to orthogonal two-color (OTC) laser field[48]: (a) PMD of TDSE; (b) PMD of TRCM; (c) PMD of Coulomb-modified strong field approximation (MSFA); (d) PMD of SFA. The horizontal white line in each figure displays the shoulder position values predicted by TRCM, the middle part of the vertical double white lines is a rectangular structure, and the horizontal red arrow indicates the width of the rectangular structure. It can be seen that the positions of the TRCM shoulder and rectangular structure are quantitatively consistent with the results of TDSE.
图 6 响应时间理论应用于分子体系[49] (a)相同激光参数下对称分子离子
$\rm H_2^ + $ 和同样电离能的模型原子势函数曲线比较, 插图是放大隧穿出点位置处两者的区别; (b)—(j) 不同激光参数条件下TDSE及TRCM理论给出的$\rm H_2^ + $ 和原子动量分布偏转角、电离时间延迟的比较Fig. 6. Application of response time theory to molecular system[49]: (a) Comparison of potential function curves of symmetric molecular ion
$\rm H_2^ + $ and model atom with the same ionization energy under the same laser parameters, the inset magnifies the difference at the tunneling exit; (b)–(j) comparison of$\rm H_2^ + $ and model atom PMD offset angles and ionization time lags given by TDSE and TRCM under different laser parameters.
图 7 响应时间理论应用于极性分子体系[52]. (a)—(c)不同方案得到的极性分子HeH+光电子动量分布结果比较 (a) TDSE; (b) MSFA-PD; (c) MSFA. (d) TRCM理论模型偏转角公式的绝热版本给出的角度-时间对应曲线
$ \theta = {\text{arctan}}[{A_x}(t)/{A_y}(t)] $ , 脉冲前后半个周期极性分子的偏转角相差2°—3°. (e)—(j) 不同激光参数下脉冲前后半个周期极性分子电离时间延迟(响应时间)及差值的比较. 前后半个周期响应时间展示了10—20 as的差异Fig. 7. Application of response time theory to polar molecular system [52]. (a)–(c) Comparison of PMD results of polar molecules HeH+ obtained from different methods: (a) TDSE; (b) MSFA-PD; (c) MSFA. (d) Corresponding curve of offset angle and time
$ \theta = {\text{arctan}}[{A_x}(t)/{A_y}(t)] $ given by the adiabatic version of the TRCM model, the offset angle of polar molecules in the first and second half laser cycle differs by 2°–3°. (e)–(j) Comparison of the ionization time lag (response time) and lag difference of polar molecules in the first and second half laser cycle under different laser parameters. The time lag of polar molecules in the first and second half laser cycle differs by 10–20 as. -
[1] Keldysh L V 1965 Sov. Phys. JETP 20 1307
[2] Ammosov M V, Delone N B, Krainov V P 1986 Sov. Phys. JETP 64 1191
[3] Schafer K J, Yang B, DiMauro L I, Kulander K C 1993 Phys. Rev. Lett. 70 1599
Google Scholar
[4] Yang B, Schafer K J, Walker B, Kulander K C, Agostini P, DiMauro L F 1993 Phys. Rev. Lett. 71 3770
Google Scholar
[5] Lewenstein M, Kulander K C, Schafer K J, Bucksbaum P H 1995 Phys. Rev. A 51 1495
Google Scholar
[6] Becker W, Grasbon F, Kopold R, Milosevic D B, Paulus G G, Walther H 2002 Adv. At. Mol. Opt. Phys. 48 35
[7] McPherson A, Gibson G, Jara H, Johann U, Luk T S, McIntyre I A, Boyer K, Rhodes C K 1987 J. Opt. Soc. Am. B 4 595
Google Scholar
[8] L’Huillier A, Schafer K J, Kulander K C 1991 J. Phys. B 24 3315
Google Scholar
[9] Corkum P B 1993 Phys. Rev. Lett. 71 1994
Google Scholar
[10] Lewenstein M, Balcou Ph, Ivanov M Yu, L’ Huillier A, Corkum P B 1994 Phys. Rev. A 49 2117
Google Scholar
[11] Niikura Hiromichi, Legare F, Hasbani R, Ivanov M Yu, Villeneuve D M, Corkum P B 2003 Nature 421 826
Google Scholar
[12] Zeidler D, Staudte A, Bardon A B, Villeneuve D M, Dörner R, Corkum P B 2005 Phys. Rev. Lett. 95 203003
Google Scholar
[13] Becker W, Liu X, Ho P J, Eberly J H 2012 Rev. Mod. Phys. 84 1011
Google Scholar
[14] Krausz F, Ivanov M 2009 Rev. Mod. Phys. 81 163
Google Scholar
[15] Lepine F, Ivanov M Yu, Vrakking M J J 2014 Nat. Photon. 8 195
Google Scholar
[16] Eckle P, Pfeiffer A N, Cirelli C, Staudte A, Dörner R, Muller H G, Buttiker M, Keller U 2008 Science 322 1525
Google Scholar
[17] Shafir D, Soifer H, Bruner B D, Dagan M, Mairesse Y, Patchkovskii S, Ivanov M Yu, Smirnova O, Dudovich N 2012 Nature 485 343
Google Scholar
[18] Dahlstrom J M, L’Huillier A, Maquet A 2012 J. Phys. B 45 183001
Google Scholar
[19] Paul P M, Toma E S, Breger P, Mullot G, Augé F, Balcou P, Muller H G, Agostini P 2001 Science 292 1689
Google Scholar
[20] Muller H G 2002 Appl. Phys. B 74 S17
[21] Leone S R, McCurdy C W, Burgdorfer J, et al. 2014 Nat. Photon. 8 162
Google Scholar
[22] Muga J G, Sala Mayato R, Egusquiza I L 2002 Time in Quantum Mechanics (Vol. 1) (Berlin, Heidelberg: Springer) pp5–6
[23] Schultze M, Fieß M, Karpowicz N, et. al. 2010 Science 328 1658
Google Scholar
[24] Saalmann U, Rost J M 2020 Phys. Rev. Lett. 125 113202
Google Scholar
[25] Landsman A S, Weger M, Maurer J, Boge R, Ludwig A, Heuser S, Cirelli C, Gallmann L, Keller U 2014 Optica 1 343
Google Scholar
[26] Sainadh U S, Xu H, Wang X, Atia-Tul-Noor A, Wallace W C, Douguet N, Bray A, Ivanov I, Bartschat K, Kheifets A, Sang R T, Litvinyuk I V 2019 Nature 568 75
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Google Scholar
[28] Boge R, Cirelli C, Landsman A S, Heuser S, Ludwig A, Maurer J, Weger M, Gallmann L, Keller U 2013 Phys. Rev. Lett. 111 103003
Google Scholar
[29] Pfeiffer A N, Cirelli C, Smolarski M, Dimitrovski D, Abu-samha M, Madsen L B, Keller U 2012 Nat. Phys. 8 76
Google Scholar
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Google Scholar
[31] Xie X J, Chen C, Xin G G, Liu J, Chen Y J 2020 Opt. Express 28 33228
Google Scholar
[32] Etches A, Madsen L B 2010 J. Phys. B 43 155602
Google Scholar
[33] Wang S, Cai J, Chen Y J 2017 Phys. Rev. A 96 043413
Google Scholar
[34] Lein M, Hay N, Velotta R, Marangos J P, Knight P L 2002 Phys. Rev. Lett. 88 183903
Google Scholar
[35] Chen Y J, Liu J, Hu Bambi 2009 Phys. Rev. A 79 033405
Google Scholar
[36] Frumker E, Hebeisen C T, Kajumba N, Bertrand J B, Wörner H J, Spanner M, Villeneuve D M, Naumov A, Corkum P B 2012 Phys. Rev. Lett. 109 113901
Google Scholar
[37] Etches A, Gaarde M B, Madsen L B 2012 Phys. Rev. A 86 023818
Google Scholar
[38] Li W Y, Yu S J, Wang S, Chen Y J 2016 Phys. Rev. A 94 053407
Google Scholar
[39] Brabec T, Ivanov M Yu, Corkum P B 1996 Phys. Rev. A 54 R2551
Google Scholar
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Google Scholar
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Google Scholar
[42] Goreslavski S P, Paulus G G, Popruzhenko S V, Shvetsov-Shilovski N I 2004 Phys. Rev. Lett. 93 233002
Google Scholar
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Google Scholar
[45] Yan T M, Popruzhenko S V, Vrakking M J J, Bauer D 2010 Phys. Rev. Lett. 105 253002
Google Scholar
[46] Che J Y, Huang J Y, Zhang F B, Chen C, Xin G G, Chen Y J 2023 Phys. Rev. A 107 043109
Google Scholar
[47] Landsman A S, Hofmann C, Pfeiffer A N, Cirelli C, Keller U 2013 Phys. Rev. Lett. 111 263001
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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