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超快超强激光及阿秒测量技术的诞生和发展, 使人们在阿秒时间和原子空间尺度内观测及控制电子的运动成为可能. 日益精密的实验测量技术对理论计算方法的精确性提出了更高的要求, 如何使用理论模型从实验结果中分辨提取超快时空动力学时间和空间信息面临极大的挑战. 相比于精确求解含时薛定谔方程, 费曼路径积分强场动力学计算方法模型简单计算效率更高, 电子波包被看作具有不同初始状态的粒子, 通过解析粒子的运动状态便能厘清各种强场非线性物理现象的产生原因. 本文从强场近似理论模型出发介绍了强场动力学计算中的鞍点近似, 进一步详细介绍了库仑修正强场近似、基于轨迹的库仑修正强场近似与库仑量子轨迹强场近似等方法. 本综述旨在为强场动力学理论计算的研究提供相关方法与文献参考, 为进一步开展新型算法提供思路.The emergence and development of ultrafast intense lasers and attosecond measurement techniques have made it possible to observe and control the motions of electrons on a timescale of attoseconds and a spatial scale of atoms. With the improvement of experimental measurement accuracy, higher requirements are put forward for the accuracy of theoretical calculation methods. Extracting temporal and spatial information about ultrafast dynamics from experimental results through using theoretical models presents a significant challenge. Compared with the exact solutions of the time-dependent Schrödinger equation, the Feynman path-integral method for strong-field dynamics calculations offers a simpler model and higher computational efficiency. The electronic wave packet is regarded as a particle with different initial states, and by analyzing the motion of the particle, the causes of various nonlinear physical phenomena in strong fields can be clarified. This work introduces the saddle point approximation into strong field dynamics calculations based on the strong field approximation theory. Furthermore, the Coulomb-corrected strong field approximation method, trajectory-based Coulomb-corrected strong field approximation method, and Coulomb quantum trajectory strong field approximation method are presented in detail. This review aims to provide relevant methods and literature references for studying strong field dynamics theoretical calculations and also to present some ideas for developing new algorithms.
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Google Scholar
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图 2 牛顿迭代法图示. 蓝色曲线为方程
$ f\left(x\right) $ 的解, 红色直线为蓝色曲线在自变量$ x $ 处的切线,$ {x}_{*} $ 为方程$ f\left(x\right)=0 $ 时需寻找的解Fig. 2. Illustration of Newton’s method. Blue curve represents value of function
$ f\left(x\right) $ , and red lines represent tangent to blue curve at independent variable$ x $ , which is solution$ {x}_{*} $ when$ f\left(x\right)=0 $ .
图 3 4个样本的
$ \left|f\right| $ 随迭代次数的变化. 蓝线、橙线、黄线和紫线分别代表初始试探解为$ ({t}_{{\rm{r}}}=20.1, {t}_{{\rm{i}}}=80.1) $ 、$ ({t}_{{\rm{r}}}= $ $ 40.1, {t}_{{\rm{i}}}=80.1) $ ,$ ({t}_{{\rm{r}}}=60.1, {t}_{{\rm{i}}}=80.1) $ 和$ ({t}_{{\rm{r}}}=120.1, {t}_{{\rm{i}}}= $ $ 80.1) $ 时, 随迭代次数增加函数值$ \left|f\right| $ 的变化.Fig. 3. Variation of
$ \left|f\right| $ with the number of iterations n for four samples. The blue, orange, yellow, and purple lines represent the changes in function values$ \left|f\right| $ with increasing iteration times when the initial trial solutions are$ ({t}_{{\rm{r}}}=20.1, {t}_{{\rm{i}}}=80.1) $ ,$ ({t}_{{\rm{r}}}=40.1, {t}_{{\rm{i}}}=80.1) $ ,$ ({t}_{{\rm{r}}}=60.1, $ $ {t}_{{\rm{i}}}= 80.1) $ and$ ({t}_{{\rm{r}}}=120.1, {t}_{{\rm{i}}}=80.1) $ , respectively.
图 4 复平面的路径积分.
$ {I}_{1} $ 描述了沿虚时间轴的积分, 步长为$ {\rm{i}}\Delta \tau $ .$ {I}_{2} $ 描述了沿实时间轴的积分, 步长为$ \Delta \tau $ Fig. 4. Path integral on complex plane.
$ {I}_{1} $ describes integration along imaginary time axis with a step size of$ {\rm{i}}\Delta \tau $ , and$ {I}_{2} $ describes the integration along real time axis with a step size of$ \Delta \tau $ .
图 5 费曼路径积分思想示意图. A与B分别为粒子的初始点与末点, 绿色虚线为粒子的可能路径 (a) 两个位置之间存在一个挡板双缝; (b) 两个位置间存在两个多缝挡板; (c) 两个位置存在无数个狭缝, 此时粒子可以从A点经历任意位置到达B点
Fig. 5. Schematic diagram of Feynman’s path integral concept. A and B represent initial and final points of a particle, and the green dashed line represents the possible paths of particle: (a) There is a double-slit barrier between two positions; (b) there are multiple slit barriers between two positions; (c) there are infinite slits between two positions, and particle can reach point B from point A through any intermediate position.
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[1] Schrödinger E 1926 Ann. Phys. 79 361
[2] Born M, Jordan P 1925 Zeit. Phys. 34 858
Google Scholar
[3] Feynman R P 1948 Rev. Mod. Phys. 20 367
Google Scholar
[4] Feynman R P, Hibbs A R 1965 Quantum Mechanics and Path Integrals (New York: McGraw Hill Press) p77
[5] Maiman T H 1960 Nature 187 493
Google Scholar
[6] Voronov G S, Delone N B 1965 JETP Lett. 1 66
[7] Agostini P, Barjot G, Bonnal J, Mainfray G, Manus C, Multiphoton J M 1968 IEEE J. Quantum Electron. 4 667
Google Scholar
[8] Agostini P, Fabre F, Mainfray G, Petite G, Rahman N K 1979 Phys. Rev. Lett. 42 1127
Google Scholar
[9] Keldysh L V 1965 Sov. Phys. JETP. 20 1307
[10] Faisal F H M 1973 J. Phys. B: At. Mol. Opt. Phys. 6 L89
[11] Reiss H R 1980 Phys. Rev. A 22 1786
Google Scholar
[12] Perelomov A M, Popov V S, Terent’ev M V 1966 Sov. Phys. JETP. 23 924
[13] Ammosov M V, Delone N B, Krainov V P 1986 Sov. Phys. JETP. 64 1191
[14] Lewenstein M, Balcou P, Ivanov M Y, L’ Huillier A, Corkum P B 1994 Phys. Rev. A 49 2117
Google Scholar
[15] Mosert V, Bauer D 2016 Comput. Phys. Commun. 207 452
Google Scholar
[16] Tao L, Scrinzi A 2012 New J. Phys. 14 013021
Google Scholar
[17] Jain M, Tzoar N 1978 Phys. Rev. A 18 538
Google Scholar
[18] Duchateau G, Cormier E, Gayet R 2002 Phys. Rev. A 66 023412
Google Scholar
[19] Yu S G, Wang Y L, Lai X Y, Huang Y Y, Quan W, Liu X J 2016 Phys. Rev. A 94 033418
Google Scholar
[20] Yudin G L, Chelkowski S, Bandrauk A D 2006 J. Phys. B: At. Mol. Opt. Phys. 39 L17
Google Scholar
[21] Popruzhenko S V, Paulus G G, Bauer D 2008 Phys. Rev. A 77 053409
Google Scholar
[22] Popruzhenko S V, Bauer D 2008 J. Mod. Optic. 55 2573
Google Scholar
[23] Yan T M, Popruzhenko S V, Vrakking M J J, Bauer D 2010 Phys. Rev. Lett. 105 253002
Google Scholar
[24] Yan T M, Bauer D 2012 Phys. Rev. A 86 053403
Google Scholar
[25] Lai X Y, Poli C, Schomerus H, Figueira de Morisson Faria C 2015 Phys. Rev. A 92 043407
Google Scholar
[26] Lai X Y, Yu S G, Huang Y Y, Hua L Q, Gong C, Quan W, Figueira de Morisson Faria C, Liu X J 2017 Phys. Rev. A 96 013414
Google Scholar
[27] Corkum P B, Burnett N H, Brunel F 1989 Phys. Rev. Lett. 62 1259
Google Scholar
[28] Salières P, L’Huillier A, Lewenstein M 1995 Phys. Rev. Lett. 74 3776
Google Scholar
[29] Salières P, Carré B, Le Déroff L, Grasbon F, Paulus G G, Walther H, Kopold R, Becker W, Milosÿevic D B 2001 Science 292 902
Google Scholar
[30] Huismans Y, Rouzée A, Gijsbertsen A, Jungmann J H, Smolkowska A S, Logan 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 U, Smirnova O, Bauer D, Porruzhenko S V, Vrakking M J J 2011 Science 331 61
Google Scholar
[31] Li M, Geng J W, Liu H, Deng Y, Wu C Y, Peng L Y, Gong Q H, Liu Y Q 2014 Phys. Rev. Lett. 112 113002
Google Scholar
[32] Hu B, Liu J, Chen S 1997 Phys. Lett. A. 236 533
Google Scholar
[33] Shvetsov-Shilovski N I, Lein M, Madsen L B, Räsänen E, Lemell C, Burgdörfer J, Arbó D G, Tókési K 2016 Phys. Rev. A 94 013415
Google Scholar
[34] Song X, Lin C, Sheng Z H, Liu P, Chen Z J, Yang W F, Hu S L, Lin C D, Chen J 2016 Sci. Rep. 6 28392
Google Scholar
[35] Liu M M, Li M, Wu C Y, Gong Q H, André Staudte, Liu Y Q 2016 Phys. Rev. Lett. 116 163004
Google Scholar
[36] Liu M M, Liu Y Q 2017 J. Phys. B: At. Mol. Opt. Phys. 50 105602
Google Scholar
[37] Gong X C, Lin C, He F, Song Q Y, Lin K, Ji Q Y, Zhang W B, Ma J Y, Lu P F, Liu Y Q, Zeng H P, Yang W F, Wu J 2017 Phys. Rev. Lett. 118 143203
Google Scholar
[38] Song X H, Shi G L, Zhang G J, Xu J W, Lin C, Chen J, Yang W F 2018 Phys. Rev. Lett. 121 103201
Google Scholar
[39] Porat G, Alon G, Rozen S, Pedatzur O, Krüger M, Azoury D, Natan A, Orenstein G, Bruner B D, Vrakking M J J, Dudovich N 2018 Nat. Commun. 9 2805
Google Scholar
[40] Trabert D, Brennecke S, Fehre K, Anders N, Geyer A, Grundmann S, Schöffler M S, Schmidt L Ph H, Jahnke T, Dörner R, Kunitski M, Eckart S 2021 Nat. Commun. 12 1697
Google Scholar
[41] Torlina L, Morales F, Kaushal J, Ivanov I, Kheifets A, Zielinski A, Scrinzi A, Muller H G, Sukiasyan S, Ivanov M Smirnova O 2015 Nat. Phys. 11 503
Google Scholar
[42] Tong J H, Liu X W, Dong W H, Jiang W Y, Zhu M, Xu Y D, Zuo Z T, Lu P F, Gong X C, Song X H, Yang W F, Wu J 2022 Phys. Rev. Lett. 129 173201
Google Scholar
[43] Scully M O, Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University Press) pp146–149
[44] Bauer D, Milošević D B, Becker W 2005 Phys. Rev. A 72 023415
Google Scholar
[45] Bleistein N, Handelsman R A 1986 Asymptotic Expansions of Integrals (Dover: Dover Publications) p252
[46] Booth A D 1949 J. Mech. Appl. Math. 2 460
Google Scholar
[47] Booth A D 1947 Nature 160 196
Google Scholar
[48] Huang L, Wu T 2018 Theor. Biol. Med. Modell. 15 22
Google Scholar
[49] Lai X Y, Figueira de Morisson Faria C 2013 Phys. Rev. A 88 013406
Google Scholar
[50] Maxwell A S 2019 Ph. D. Dissertation (London: University College London
[51] Yang S D, Song X H, Liu X W, Zhang H D, Shi G L, Yu X H, Tang Y J, Chen J, Yang W F 2020 Laser Phys. Lett. 17 095301
Google Scholar
[52] Becker W, Grasbon F, Kopold R, Milošević D B, Paulus G G, Walther H 2002 Adv. At. Mol. Opt. Phys. 48 35
Google Scholar
[53] Paul M, Gräfe S 2019 Phys. Rev. A 99 053414
Google Scholar
[54] Liu M M, Shao Y, Han M, Ge P P, Deng Y K, Wu C Y, Gong Q H, Liu Y Q 2018 Phys. Rev. Lett. 120 043201
Google Scholar
[55] Yan T M 2012 Ph. D. Dissertation (Germany: Universität Rostock
[56] Arbó D G, Ishikawa K L, Persson E, Burgdörfer J 2012 Nucl. Instrum. Methods Phys. Res. , Sect. B 279 24
Google Scholar
[57] Milošević D B, Paulus G G, Bauer D, Becker W 2006 J. Phys. B: At. Mol. Opt. Phys. 39 R203
Google Scholar
[58] Becker A, Faisal F H M 2005 J. Phys. B: At. Mol. Opt. Phys. 38 R1
Google Scholar
[59] Popov V S 2004 Phys. Usp. 47 855
Google Scholar
[60] Xiao X R, Wang M X, Xiong W H, Peng L Y 2016 Phys. Rev. E 94 053310
Google Scholar
[61] 李庆扬, 王能超, 易大义 2008 数值分析(第五版) (北京: 清华大学出版社) 第286页
Li Q Y, Wang N C, Yi D Y 2008 Numerical Analysis (5th Ed.) (Beijing: Tsinghua University Press) p286
[62] Shvetsov-Shilovski N I, Goreslavski S P, Popruzhenko S V, Becker W 2009 Laser Phys. 19 1550
Google Scholar
[63] Shvetsov-Shilovski N I, Dimitrovski D, Madsen L B 2012 Phys. Rev. A 85 023428
Google Scholar
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