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Purpose: The interaction of intense, ultrashort laser pulses with atoms gives rise to rich non-perturbative phenomena, which are encoded within the final-state photoelectron momentum distribution (PMD). A particularly enigmatic feature often observed in the multiphoton ionization regime (Keldysh parameter $ \gamma \gtrsim 1 $), is a complex, fan-like interference pattern in the near-threshold, low-energy region of the PMD. The physical origin of this structure has been a subject of extensive debate, with proposed mechanisms ranging from multipath interference in the Coulomb field to complex sub-barrier dynamics. This work aims to provide a physical explanation for this phenomenon. We hypothesize and demonstrate that this fan-like structure is not only the consequence of Coulomb focusing, but also a direct and sensitive signature of non-adiabatic dynamics occurring as an electron tunnels through the laser-dressed atomic potential barrier. Our goal is to clearly separate the key physical ingredients responsible for shaping this quantum interference. Methodology: To achieve this, we employ a synergistic three-pronged approach that combines experiment, exact numerical simulation, and a sophisticated theoretical model. 1. Experiment: We perform velocity-map imaging measurements on argon atoms ionized by a 798-nm 35-fs laser pulse at a peak intensity of $ 6.3 \times 10^{13} $ W/cm2, and the experimental results clearly show the low-energy fan-like interference pattern. 2. Quantum Benchmark: We solve the time-dependent Schrödinger equation (TDSE) within the single-active-electron (SAE) approximation by using a well-established model potential for argon, which accurately reproduces its ionization potential and ground-state properties. After performing a focal-volume average to simulate experimental conditions, the TDSE results show excellent qualitative agreement with the measurements, establishing the TDSE as a reliable quantum benchmark for our investigation. 3. Semiclassical Model (CTMC-p): The core of our analysis relies on a custom-developed semiclassical trajectory model based on the Feynman path-integral formulation. In this framework, ionization process is divided into two steps: (i) an electron tunnels through the potential barrier at an initial time $ t_0 $ and position $ {\boldsymbol{r}}_0 $, and (ii) it propagates classically in the combined laser and ionic fields according to Newton's equations. Crucially, each trajectory is endowed with a quantum phase accumulated along its path, $ \Phi_k $, allowing for the coherent summation of all trajectories ending with the same final momentum, $ M_j = \displaystyle\sum\nolimits_k {\mathrm{e}}^{{\mathrm{i}}\Phi_k} $. Our model combines two critical physical effects beyond standard treatments: • Non-Adiabatic Tunneling: We introduce a non-zero initial longitudinal momentum, $ v_{0\parallel} =-A(t_0)\times $$ \left(\sqrt{1+\gamma_{\text{eff}}^2}-1\right) $, acquired by the electron at the tunnel exit. This term accounts for the non-instantaneous nature of the tunneling process, a key non-adiabatic effect. • Core Polarization: We include an induced dipole potential, $ U_{\text{ID}} = -\alpha^I {\boldsymbol{E}}(t) \cdot {\boldsymbol{r}}/r^3 $, to model the dynamic polarization of the Ar$ ^+ $ ionic core, a multi-electron effect. By selectively including or excluding these effects, we can clearly isolate their respective contributions to the final PMD. Results: Our central finding is that the non-adiabatic initial longitudinal momentum is the decisive factor for correctly describing the near-threshold interference. This is powerfully illustrated in Figure 6 . The benchmark TDSE calculation [Fig. 6(a) ] for a single intensity of $ 5 \times 10^{13} $ W/cm2 ($ \gamma \approx 1.6 $) reveals a distinct 6-lobe interference pattern. A traditional semiclassical simulation based on the quasi-static tunneling approximation (i.e., setting $ v_{0\parallel} = 0 $) qualitatively fails, predicting an incorrect 8-lobe structure [Fig. 6(c) ]. However, upon including the non-zero initial longitudinal momentum ($ v_{0\parallel} \neq 0 $), our non-adiabatic semiclassical model quantitatively reproduces the correct 6-lobe structure, showing that it is in excellent agreement with the TDSE benchmark [Fig. 6(b) ].To understand the underlying physics, we perform a quantum-orbit decomposition. This analysis reveals that the overall fan-like structure arises from the interference of multiple trajectory types, including ‘direct’ (Category I), ‘forward-scattered’ (Category Ⅱ, and ‘glory-scattered’ (Category Ⅲ) orbits. Although the entire structure arises from the collective interference of these paths, we pinpoint the origin of the lobe-count correction. The initial longitudinal momentum contributes a phase term, $ \Delta\Phi_{\text{initial}} \approx $$ -{\boldsymbol{v}}_{0\parallel} \cdot {\boldsymbol{r}}_0 $, to the total accumulated action. We find that the relative phase between the ‘direct’ and ‘glory’ trajectories is exquisitely sensitive to this term due to their vastly different paths and birth conditions. It is this specific and dramatic change in the Ⅰ–Ⅲ interference channel that ultimately corrects the topology of the entire pattern, reducing the lobe count from 8 to 6. In contrast, other interference pairs, such as the holographic pair II-III, are largely robust against this effect as their nearly identical birth conditions cause the initial phase term to cancel out in their relative phase. In parallel, our simulations show that the ionic core polarization has a negligible effect on this low-energy structure, but is essential for accurately describing higher-energy rescattering features by smoothing unphysical caustics caused by a pure Coulomb potential. Conclusion: We demonstrate clearly that the near-threshold fan-like interference pattern in the multiphoton regime is a direct manifestation of non-adiabatic dynamics during tunneling, specifically the acquisition of a longitudinal momentum component by the electron during its finite-time passage under the potential barrier. Our findings not only provide a clear, intuitive, and orbit-based physical picture for this complex quantum phenomenon but also highlight the predictive power of semiclassical methods when crucial non-adiabatic effects are properly incorporated. This understanding lays a foundation for future investigations, including the extension of this model to more complex molecular systems and its application in retrieving attosecond electron dynamics from holographic interference patterns. -
Keywords:
- non-adiabatic tunneling dynamics /
- quantum orbit interference /
- semiclassical method /
- near-threshold photoelectron spectra
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图 1 (a)实验与(b)含时薛定谔方程模拟的氩原子光电子动量分布(PMD)对比(激光参数: 798 nm, 35 fs, 强度$ 6.3 \times 10^{13} $ W/cm2). 激光偏振沿$ p_\parallel $轴. (c) 对应光电子能谱(全发射角积分): 实验(绿线)与含时薛定谔方程模拟(蓝线). 含时薛定谔方程结果已作焦斑体积平均
Figure 1. Comparison of (a) experimental and (b) TDSE-simulated photoelectron momentum distributions (PMDs) for argon ionized by a 798 nm, 35 fs laser pulse with a laser intensity of $ 6.3 \times 10^{13} $ W/cm2. The laser polarization is along the $ p_\parallel $ axis. (c) The corresponding photoelectron energy spectra, integrated over all emission angles, for experiment (green dotted line) and TDSE (blue dotted line). The TDSE results have been focal-volume averaged.
图 2 初始纵向动量的关键作用 (a) 单强度含时薛定谔方程模拟基准PMD ($ I = 5 \times 10^{13} $ W/cm2, $ \gamma \approx 1.6 $). (b) 含与 (c) 不含隧穿出口初始纵向动量的半经典模拟低能PMD对比. 含初始动量的模拟(b)正确复现了TDSE基准(a)的6瓣结构, 而未包含的模拟(c)错误预测了8瓣
Figure 2. The critical role of initial longitudinal momentum. (a) Benchmark PMD from a single-intensity TDSE simulation ($ I = 5 \times 10^{13} $ W/cm2, $ \gamma \approx 1.6 $). (b, c) Comparison of low-energy PMDs from semiclassical simulations with (b) and without (c) the inclusion of the initial longitudinal momentum at the tunnel exit. The simulation including the initial momentum (b) correctly reproduces the 6-lobe structure seen in the TDSE benchmark (a), while the simulation without it (c) incorrectly predicts 8 lobes.
图 3 干涉机制解析. 仅Ⅰ+Ⅱ类(“直接”与前向散射)(a, b)和Ⅱ+Ⅲ类(全息对)(c, d)干涉的PMD. (a)(c) 含初始纵向动量, (b)(d) 不含. Ⅰ+Ⅱ类干涉的扇形结构基本不受初始动量影响, 而主导SFPH的Ⅱ+Ⅲ类干涉同样稳健
Figure 3. Dissection of interference mechanisms. PMDs showing interference between only Categories Ⅰ and Ⅱ (“direct” and forward-scattered) (a, b) and Categories Ⅱ and Ⅲ (holographic pair) (c, d). Panels (a) and (c) include the initial longitudinal momentum, while (b) and (d) do not. The fanlike structure from Ⅰ+Ⅱ interference is largely unaffected by the initial momentum, whereas the Ⅱ+Ⅲ interference, responsible for SFPH, is also robust.
图 4 修正后干涉图案的物理起源. 仅Ⅰ类(“直接”)与Ⅲ类(“glory”散射)轨迹干涉的PMD: (a)含与(b)不含初始纵向动量. 显著变化可见: 含初始动量时(a)在约$ 30^\circ $处正确产生干涉极小(错误模型(b)在此形成明显凸瓣). 此特定修正改变了完整PMD的总瓣数
Figure 4. The physical origin of the corrected interference pattern. PMDs arising from the interference of only Category Ⅰ (“direct”) and Category Ⅲ (“glory” scattered) trajectories, calculated (a) with and (b) without the initial longitudinal momentum. A dramatic change is visible: including the initial momentum (a) correctly creates an interference minimum along the angle where the incorrect model (b) produced a prominent lobe (approx. $ 30^\circ $). This specific change is what corrects the overall lobe count in the full PMD.
图 5 感应偶极势对较高能再散射结构的影响 (a) 含时薛定谔方程(TDSE)模拟(图2(a)放大版). (b) 不含感应偶极势的半经典模拟($ v_{0\parallel}\ne 0,\; U_{{\mathrm{ID}}} = 0 $)(图2(c)放大版). (c) 包含感应偶极势的非绝热半经典模拟($ v_{0\parallel}\neq 0,\; U_{{\mathrm{ID}}}\neq 0 $). 对比可见, 引入$ U_{{\mathrm{ID}}} $显著抑制了纯库仑势导致的尖锐焦散结构, 使结果(c)在定性上更接近TDSE基准(a)
Figure 5. The role of the induced dipole potential on high-energy rescattering structures. (a) The TDSE simulation (zoom-in of Fig. 2(a)). (b) The semiclassical simulation without induced dipole potential ($ v_{0\parallel}\ne 0,\; U_{{\mathrm{ID}}} = 0 $) (zoom-in of Fig. 2(c)). (c) Result from the non-adiabatic semiclassical simulation including the induced dipole potential ($ v_{0\parallel}\neq 0,\; U_{{\mathrm{ID}}}\neq 0 $). The comparison shows that including $ U_{{\mathrm{ID}}} $ significantly suppresses the sharp caustics caused by the pure Coulomb potential, making the result in (c) qualitatively closer to the TDSE benchmark in (a).
图 6 The critical role of initial longitudinal momentum. (a) Benchmark PMD from a single-intensity TDSE simulation ($ I = 5\times10^{13} $ W/cm2, $ \gamma \approx 1.6 $). (b, c) Comparison of low-energy PMDs from semiclassical simulations with (b) and without (c) the inclusion of the initial longitudinal momentum at the tunnel exit. The simulation including the initial momentum (b) correctly reproduces the 6-lobe structure seen in the TDSE benchmark (a), while the simulation without it (c) incorrectly predicts 8 lobes.
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[1] Krausz F, Ivanov M 2009 Rev. Mod. Phys. 81 163
Google Scholar
[2] Pazourek R, Nagele S, Burgdörfer J 2015 Rev. Mod. Phys. 87 765
Google Scholar
[3] Agostini P, Fabre F, Mainfray G, Petite G, Rahman N K 1979 Phys. Rev. Lett. 42 1127
Google Scholar
[4] Paulus G G, Nicklich W, Xu H, Lambropoulos P, Walther H 1994 Phys. Rev. Lett. 72 2851
Google Scholar
[5] Krause J L, Schafer K J, Kulander K C 1992 Phys. Rev. Lett. 68 3535
Google Scholar
[6] Walker B, Sheehy B, DiMauro L F, Agostini P, Schafer K J, Kulander K C 1994 Phys. Rev. Lett. 73 1227
Google Scholar
[7] Huismans Y, Rouzée A, Gijsbertsen A, Jungmann J H, Smolkowska A S, Logman P S W M, Lépine F, Cauchy C, Zamith S, Marchenko T, Bakker J M, Berden G, Redlich B, van der Meer A F G, Muller H G, Vermin W, Schafer K J, Spanner M, Ivanov M Y, Smirnova O, Bauer D, Popruzhenko S V, Vrakking M J J 2011 Science 331 61
Google Scholar
[8] He M, Li Y, Zhou Y, Li M, Cao W, Lu P 2018 Phys. Rev. Lett. 120 133204
Google Scholar
[9] Xie W, Yan J, Li M, Cao C, Guo K, Zhou Y, Lu P 2021 Phys. Rev. Lett. 127 263202
Google Scholar
[10] Li M, Xie H, Cao W, Luo S, Tan J, Feng Y, Du B, Zhang W, Li Y, Zhang Q, Lan P, Zhou Y, Lu P 2019 Phys. Rev. Lett. 122 183202
Google Scholar
[11] Tao J F, Cai J, Xia Q Z, Liu J 2020 Phys. Rev. A 101 043416
Google Scholar
[12] 陶建飞, 夏勤智, 廖临谷, 刘杰, 刘小井 2022 71 233206
Google Scholar
Tao J F, Xia Q Z, Liao L G, Liu J, Liu X J 2022 Acta Phys. Sin. 71 233206
Google Scholar
[13] 黄雪飞, 苏杰, 廖健颖, 李盈傧, 黄诚 2022 71 093202
Google Scholar
Huang X F, Su J, Liao J Y, Li Y B, Huang C 2022 Acta Phys. Sin. 71 093202
Google Scholar
[14] He M, Fan Y, Zhou Y, Lu P 2021 Chinese Phys. B 30 123202
Google Scholar
[15] Muller H G 1999 Phys. Rev. A 60 1341
Google Scholar
[16] HuP B, Liu J, gang Chen S 1997 Phys. Lett. A 236 533
Google Scholar
[17] Liu J, Xia Q Z, Tao J F, Fu L B 2013 Phys. Rev. A 87 041403
Google Scholar
[18] Ding B, Xu W, Wu R, Feng Y, Tian L, Li X, Huang J, Liu Z, Liu X 2021 Appl. Sci. 1 1
[19] Hickstein D D, Gibson S T, Yurchak R, Das D D, Ryazanov M 2019 Rev. Sci. Instrum. 90 065115
Google Scholar
[20] Tulsky V, Bauer D 2020 Comput. Phys. Comm. 251 107098
Google Scholar
[21] Tong X M, Lin C D 2005 J. Phys. B: At. Mol. Opt. Phys. 38 2593
Google Scholar
[22] Tao L, Scrinzi A 2012 New J. Phys. 14 013021
Google Scholar
[23] Shvetsov-Shilovski N I, Lein M, Madsen L B, Räsänen E, Lemell C, Burgdörfer J, Arbó D G, Tőkési K 2016 Phys. Rev. A 94 013415
Google Scholar
[24] Delone N B, Krainov V P 1991 J. Opt. Soc. Am. B 8 1207
[25] Arissian L, Smeenk C, Turner F, Trallero C, Sokolov A V, Villeneuve D M, Staudte A, Corkum P B 2010 Phys. Rev. Lett. 105 133002
Google Scholar
[26] Dreissigacker I, Lein M 2013 Chem. Phys. 414 69
Google Scholar
[27] Shvetsov-Shilovski N I, Dimitrovski D, Madsen L B 2012 Phys. Rev. A 85 023428
Google Scholar
[28] Dimitrovski D, Maurer J, Stapelfeldt H, Madsen L B 2014 Phys. Rev. Lett. 113 103005
Google Scholar
[29] Kang H P, Xu S P, Wang Y L, Yu S G, Zhao X Y, Hao X L, Lai X Y, Pfeifer T, Liu X J, Chen J, Cheng Y, Xu Z Z 2018 J. Phys. B: At. Mol. Opt. Phys 51 105601
Google Scholar
[30] Etches A, Madsen L B 2010 J. Phys. B: At. Mol. Opt. Phys 43 155602
Google Scholar
[31] Bristow M P F, Glass I I 1972 Phys. Fluids 15 2066
Google Scholar
[32] Li M, Geng J W, Han M, Liu M M, Peng L Y, Gong Q, Liu Y 2016 Phys. Rev. A 93 013402
Google Scholar
[33] Tao J F, Xia Q Z, Cai J, Fu L B, Liu J 2017 Phys. Rev. A 95 011402
Google Scholar
[34] Xia Q Z, Tao J F, Cai J, Fu L B, Liu J 2018 Phys. Rev. Lett. 121 143201
Google Scholar
[35] Liao L G, Xia Q Z, Cai J, Liu J 2022 Phys. Rev. A 105 053115
Google Scholar
[36] Wang T, Dube Z, Mi Y, Vampa G, Villeneuve D M, Corkum P B, Liu X, Staudte A 2022 Phys. Rev. A 106 013106
Google Scholar
[37] Möller M, Meyer F, Sayler A M, Paulus G G, Kling M F, Schmidt B E, Becker W, Milošević D B 2014 Phys. Rev. A 90 023412
Google Scholar
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