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拉伸到大核间距的分子离子谐波辐射谱上复杂干涉结构的物理起源

李卫艳 刘娜 王赏

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拉伸到大核间距的分子离子谐波辐射谱上复杂干涉结构的物理起源

李卫艳, 刘娜, 王赏

Physical origins of complex interference structures in harmonic emission from molecular ions stretched to large internuclear distances

Li Wei-Yan, Liu Na, Wang Shang
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  • 通过数值和解析的方法研究了拉伸到大核间距的分子离子的高次谐波辐射, 重点研究了与短轨道贡献相关的谐波辐射谱的精细结构. 研究结果表明, 大核间距分子离子的短轨道谐波辐射谱呈现了复杂的干涉结构. 通过利用一个考虑了电荷共振效应的强场近似模型, 可以分辨这些复杂的干涉结构的物理起源: 谐波辐射谱上一些干涉最小来源于电离过程中的两中心干涉效应, 而另外一些干涉最小来源于再结合过程中的两中心干涉效应, 电离干涉最小与再结合干涉最小交替出现. 与电离过程不同, 再结合过程中, 库仑加速会显著改变再结合电子的德布罗意波长, 从而改变干涉最小的位置.
    High-order harmonic generation (HHG) from the molecular ions stretched to large internuclear distances is studied numerically and analytically in this work. We focus on the fine structure of the HHG spectrum related to the contribution of short electron trajectory. In our numerically solving the time-dependent Schrodinger equation (TDSE), we use a trajectory-dependent filtering procedure to separate the short-trajectory contribution from other contributions of long trajectory and multiple returns. Our TDSE results reveal that the short-trajectory HHG spectra of molecular ion with larger internuclear distance show some complex interference structures characterized by some remarkable dips, and that the position of the dip is sensitive to the laser parameters. With a developed model arising from strong-field approximation (SFA), we are able to identify the physical origins of the complex interference structures. In this model considered is the charge-resonance effect which induces the strong coupling between the ground state and the first excited state of the molecular ion at large internuclear distance. In this model, the well-known effect of two-center interference occurs in the form of the canonical momentum instead of the momentum related to the instantaneous velocity of the electron in the general SFA. It is shown that some dips in TDSE results arise from two-center interference of the electronic wave between these two atomic cores of the molecule in the ionization process, while others come from that in the recombination process. These ionization and recombination dips alternately appear in the HHG spectra from the formed complex interference structures. The main differences between the interference effects in the ionization process and the recombination process are twofold. Firstly, in the ionization process, the destructive two-center interference strongly suppresses the forming of the continuum wavepacket, while in the recombination process, the recombination of the rescattering electron with other bound eigenstates with small weights can also contribute to HHG bedsides the recombination of the ground state with the first excited state with large weights. As a result, in TDSE results, the ionization dips are deeper and more remarkable than the recombination ones. Secondly, in the recombination process, the Coulomb acceleration remarkably changes the de Broglie wavelength of the rescattering electron and therefore changes the position of the interference-induced dip. While in the ionization process, the Coulomb potential plays a small role in the interference effect. As a result, the interference dips in the ionization process and the recombination process differ from each other.
      通信作者: 李卫艳, liwy157@126.com ; 王赏, phywangshang@163.com
    • 基金项目: 国家自然科学基金(批准号: 11904072)、河北省教育厅青年基金(批准号: QN2019074, QN2022143)和河北省省级科技计划(批准号: A2022205002)资助的课题.
      Corresponding author: Li Wei-Yan, liwy157@126.com ; Wang Shang, phywangshang@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11904072), the Science and Technology Project of Hebei Education Department for Young Scholar, China (Grant Nos. QN2019074, QN2022143), and the S&T Program of Hebei, China (Grant No. A2022205002).
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  • 图 1  $R = 16$的H$_2^ + $的短轨道TDSE谐波谱与模型预测结果, 激光场强度$ I = 5 \times {10^{14}}{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{W/c}}{{\text{m}}^2} $, 激光波长$\lambda = 1600{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{nm}}$ (a), (c) $\theta = {0^ \circ }$$\theta = {20^ \circ }$${\text{H}}_2^ + $的短轨道TDSE谐波谱; (b), (d) (5)式在$\theta = {0^ \circ }$$\theta = {20^ \circ }$ 的短轨道计算结果, 包括考虑电离干涉和再结合干涉的结果(黑色实线)和仅考虑电离干涉的结果(红色虚线). 红色和蓝色的数字分别表示电离干涉和再结合干涉最小的位置. 短箭头指示TDSE结果中一些干涉最小的位置, 竖直虚线指示(5)式预期的干涉最小的位置

    Fig. 1.  Harmonic spectra of H$_2^ + $ with $R = 16$, obtained for short-trajectory TDSE simulations and relevant predictions of Eq. (5), laser parameters used are $I = 5 \times {10^{14}} ~$W/cm2 and $\lambda = 1600$ nm: (a), (c) Short-trajectory TDSE simulations at $\theta = {0^ \circ }$and$\theta = {20^ \circ }$, respectively; (b), (d) relevant predictions of Eq. (5) at $\theta = {0^ \circ }$ and $\theta = {20^ \circ }$, respectively, with considering the contributions of both ionization and recombination (black solid curves) and considering only the contributions of ionization (red dashed curves). The red and blue numbers indicate minima related to ionization and recombination, respectively. Arrows indicate some minima in TDSE results and use the vertical dashed lines to indicate minima in predictions of Eq. (5).

    图 2  核间距$R = 16$${\text{H}}_2^ + $ 的短轨道TDSE谐波在 $\theta = {0^ \circ }$ 的时频分布以及相应(5)式预期的干涉最小, 包括联系于电离的干涉最小(红色空心圆圈)和联系于再结合的干涉最小(黑色空心圆圈), 激光参数同图1(a)

    Fig. 2.  Time-frequency distribution of HHG for ${\text{H}}_2^ + $ with $R = 16$ obtained with short-trajectory TDSE simulations at $\theta = {0^ \circ }$ and the corresponding predictions of Eq. (5) for the interference minima, including minima related to ionization (red hollow circle) and recombination (black hollow circle), respectively. The laser parameters used are as in Fig. 1(a).

    图 3  $R = 16$的H$_2^ + $的短轨道TDSE谐波谱与模型预测结果, 激光场强度$ I = 5 \times {10^{14}}{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{W/c}}{{\text{m}}^2} $, 激光波长$\lambda = 1800{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{nm}}$(a), (c)$\theta = {0^ \circ }$$\theta = {20^ \circ }$${\text{H}}_2^ + $的短轨道TDSE谐波谱; (b), (d) (5)式在$\theta = {0^ \circ }$$\theta = {20^ \circ }$ 的短轨道计算结果, 包括考虑电离干涉和再结合干涉的结果(黑色实线)和仅考虑电离干涉的结果(红色虚线). 红色和蓝色的数字分别表示电离干涉和再结合干涉的最小位置. 短箭头指示TDSE结果中一些干涉最小的位置, 竖直虚线指示(5)式预期的干涉最小的位置

    Fig. 3.  Harmonic spectra of H$_2^ + $ with $R = 16$, obtained for short-trajectory TDSE simulations and relevant predictions of Eq. (5), laser parameters used are $I = 5 \times {10^{14}}$ W/cm2 and $\lambda = 1800$ nm: (a), (c) Short-trajectory TDSE simulations at $\theta = {0^ \circ }$ and $\theta = {20^ \circ }$, respectively; (b), (d) relevant predictions of Eq. (5) at $\theta = {0^ \circ }$ and $\theta = {20^ \circ }$, respectively, with considering the contributions of both ionization and recombination (black solid curves) and considering only the contributions of ionization (red dashed curves). The red and blue numbers indicate minima related to ionization and recombination, respectively. Arrows indicate some minima in TDSE results and use the vertical dashed lines to indicate minima in predictions of Eq. (5).

    图 4  $R = 16$的H$_2^ + $的短轨道TDSE谐波谱与模型预测结果, 激光场强度$ I = 5 \times {10^{14}}{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{W/c}}{{\text{m}}^2} $ , 激光波长$\lambda = 1200{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{nm}}$(a), (c)$\theta = {0^ \circ }$$\theta = {20^ \circ }$${\text{H}}_2^ + $的短轨道TDSE谐波谱; (b), (d) (5)式在$\theta = {0^ \circ }$$\theta = {20^ \circ }$ 的短轨道计算结果, 包括考虑电离干涉和再结合干涉的结果(黑色实线)和仅考虑电离干涉的结果(红色虚线). 红色和蓝色的数字分别表示电离干涉和再结合干涉的最小位置. 短箭头指示TDSE结果中一些干涉最小的位置, 竖直虚线指示(5)式预期的干涉最小的位置

    Fig. 4.  Harmonic spectra of H$_2^ + $ with $R = 16$, obtained for short-trajectory TDSE simulations and relevant predictions of Eq. (5), laser parameters used are $I = 5 \times {10^{14}}$W/cm2 and $\lambda = 1200$ nm: (a), (c) Short-trajectory TDSE simulations at $\theta = {0^ \circ }$ and $\theta = {20^ \circ }$, respectively; (b), (d)relevant predictions of Eq. (5) at $\theta = {0^ \circ }$ and $\theta = {20^ \circ }$, respectively, with considering the contributions of both ionization and recombination (black solid curves) and considering only the contributions of ionization (red dashed curves). The red and blue numbers indicate minima related to ionization and recombination, respectively. Arrows indicate some minima in TDSE results and use the vertical dashed lines to indicate minima in predictions of Eq. (5).

    图 5  $R = 12$的H$_2^ + $的短轨道TDSE谐波谱与模型预测结果, 激光场强度$ I = 5 \times {10^{14}}{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{W/c}}{{\text{m}}^2} $, 激光波长$\lambda = 1600{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{nm}}$(a), (c)$\theta = {0^ \circ }$$\theta = {20^ \circ }$${\text{H}}_2^ + $的短轨道TDSE谐波谱; (b), (d) (5)式在$\theta = {0^ \circ }$$\theta = {20^ \circ }$ 的短轨道计算结果, 包括考虑电离干涉和再结合干涉的结果(黑色实线)和仅考虑电离干涉的结果(红色虚线). 红色和蓝色的数字分别表示电离干涉和再结合干涉的最小位置. 短箭头指示TDSE结果中一些干涉最小的位置, 竖直虚线指示(5)式预期的干涉最小的位置

    Fig. 5.  Harmonic spectra of H$_2^ + $ with $R = 12$, obtained for short-trajectory TDSE simulations and relevant predictions of Eq. (5), laser parameters used are $I = 5 \times {10^{14}}$W/cm2 and $\lambda = 1600$ nm: (a), (c) Short-trajectory TDSE simulations at $\theta = {0^ \circ }$ and$\theta = {20^ \circ }$, respectively; (b), (d) relevant predictions of Eq. (5) at $\theta = {0^ \circ }$ and $\theta = {20^ \circ }$, respectively, with considering the contributions of both ionization and recombination (black solid curves) and considering only the contributions of ionization (red dashed curves). The red and blue numbers indicate minima related to ionization and recombination, respectively. Arrows indicate some minima in TDSE results and use the vertical dashed lines to indicate minima in predictions of Eq. (5).

    图 6  核间距$R = 16$${\text{H}}_2^ + $ 的短轨道TDSE谐波在 $\theta = {0^ \circ }$ 的时频分布以及相应(5)式预期的干涉最小, 包括联系于电离的干涉最小(红色空心圆圈)和联系于再结合的干涉最小(黑色空心圆圈). 其中, 电离能${I_p} = 0.562\;{\text{a}}{\text{.u}}.$. 激光场强度$ I = 1 \times {10^{14}}{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{W/c}}{{\text{m}}^2} $, 激光波长$\lambda = 1800{\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\kern 1 pt} {\text{nm}}$

    Fig. 6.  Time-frequency distribution of HHG for ${\text{H}}_2^ + $ with $R = 16$ obtained with short-trajectory TDSE simulations at $\theta = {0^ \circ }$ and the corresponding predictions of Eq. (5) for the interference minima, including minima related to ionization (red hollow circle) and recombination (black hollow circle), respectively. Ionization energy${I_p} = 0.562\;{\text{a}}{\text{.u}}.$. Laser parameters used are $I = 1 \times {10^{14}}$W/cm2 and $\lambda = 1800$ nm

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    Tao Z, Chen C, Szilvasi T, Keller M, Mavrikakis M, Kapteyn H, Murnane M 2016 Science 353 62Google Scholar

    [3]

    Neutze R, Wouts R, van der Spoel D, Weckert E, Hajdu J 2000 Nature 406 752Google Scholar

    [4]

    Hentschel M, Kienberger R, Spielmann Ch, Reider G A, Milosevic N, Brabec T, Corkum P, Heinzmann U, Drescher M, Krausz F 2001 Nature 414 509Google Scholar

    [5]

    Ihee H, Lobastov V A, Gomez U M, Goodson B M, Srinivasan R, Ruan C Y, Zewail A H 2001 Science 291 458Google Scholar

    [6]

    Baltuska A, Udem Th, Uiberacker M, Hentschel M, Goulielmakis E, Gohle Ch, Holzwarth R, Yakovlev V S, Scrinzi A, Hänsch T W, Krausz F 2003 Nature 421 611Google Scholar

    [7]

    Eckle P, Smolarski M, Schlup F, Biegert J, Staudte A, Schöffler M, Muller H G, Dörner R, Keller U 2008 Nat. Phys. 4 565Google Scholar

    [8]

    Corkum P B 1993 Phys. Rev. Lett. 71 1994Google Scholar

    [9]

    Itatani J, Zeidler D, Levesque J, Spanner M, Villeneuve D M, Corkum P B 2005 Phys. Rev. Lett. 94 123902Google Scholar

    [10]

    Le V H, Le A T, Xie R H, Lin C D 2007 Phys. Rev. A 76 013414Google Scholar

    [11]

    McFarland B K, Farrell J P, Bucksbaum P H, Gühr M 2008 Science 322 1232Google Scholar

    [12]

    Shafir D, Mairesse Y, Villeneuve D M, Corkum P B, Dudovich N 2009 Nat. Phys. 5 412Google Scholar

    [13]

    Vozzi C, Negro M, Calegari F, Sansone G, Nisoli M, De Silvestri S, Stagira S 2011 Nat. Phys. 7 822Google Scholar

    [14]

    Lein M, Hay N, Velotta R, Marangos J R, Knight P L 2002 Phys. Rev. Lett. 88 183903Google Scholar

    [15]

    Kanai T, Minemoto S, Sakai H 2005 Nature 435 470Google Scholar

    [16]

    Vozzi C, Calegari F, Benedetti E, Caumes J P, Sansone G, Stagira S, Nisoli M, Torres R, Heesel E, Kajumba N, Marangos J P, Altucci C, Velotta R 2005 Phys. Rev. Lett. 95 153902Google Scholar

    [17]

    Chen Y J, Liu J, Hu Bambi 2009 J. Chem. Phys. 130 044311Google Scholar

    [18]

    Smirnova O, Mairesse Y, Patchkovskii S, Dudovich N, Villeneuve D, Corkum P, Ivanov M Yu 2009 Nature 460 972Google Scholar

    [19]

    Wörner H J, Bertrand J B, Kartashov D V, Corkum P B, Villeneuve D M 2010 Nature 466 604Google Scholar

    [20]

    Salières P, Maquet A, Haessler S, Caillat J, Taïeb R 2012 Rep. Prog. Phys. 75 062401Google Scholar

    [21]

    Lépine M, Ivanov M Yu, Vrakking M J J 2014 Nat. Photon. 8 195Google Scholar

    [22]

    Shafir D, Soifer H, Bruner B D, Dagan M, Mairesse Y, Patchkovskii S, Ivanov M Yu, Smirnova O, Dudovich N 2012 Nature 485 343Google Scholar

    [23]

    Zhao J, Lein M 2013 Phys. Rev. Lett. 111 043901Google Scholar

    [24]

    Etches A, Madsen L B 2010 J. Phys. B 43 155602Google Scholar

    [25]

    Frumker E, Kajumba N, Bertrand J B, Wörner H J, Hebeisen C T, Hockett P, Spanner M, Patchkovskii S, Paulus G G, Villeneuve D M, Naumov A, Corkum P B 2012 Phys. Rev. Lett. 109 233904Google Scholar

    [26]

    Kraus P M, Rupenyan A, Wörner H J 2012 Phys. Rev. Lett. 109 233903Google Scholar

    [27]

    Kraus P M, Baykusheva D, Wörner H J 2014 Phys. Rev. Lett. 113 023001Google Scholar

    [28]

    Lewenstein M, Balcou Ph, Ivanov M Yu, L’Huillier A, Corkum P B 1994 Phys. Rev. A 49 2117Google Scholar

    [29]

    Zuo T, Bandrauk A D 1995 Phys. Rev. A 52 R2511Google Scholar

    [30]

    Becker W, Chen J, Chen S G, Milošević D B 2007 Phys. Rev. A 76 033403Google Scholar

    [31]

    Chen Y J, Zhang B 2012 J. Phys. B 45 215601Google Scholar

    [32]

    Zuo T, Chelkowski S, Bandrauk A D 1993 Phys. Rev. A 48 3837Google Scholar

    [33]

    Ivanov M Yu, Corkum P B 1993 Phys. Rev. A 48 580Google Scholar

    [34]

    Chen Y J, Chen J, Liu J 2006 Phys. Rev. A 74 063405Google Scholar

    [35]

    Chen J, Chen S G 2007 Phys. Rev. A 75 041402(RGoogle Scholar

    [36]

    Li W Y, Dong F L, Yu S J, Wang S, Yang S P, Chen Y J 2015 Opt. Express 23 31010Google Scholar

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    Li W Y, Wang S, Shi Y Z, Yang S P, Chen Y J 2017 J. Phys. B 50 085003Google Scholar

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出版历程
  • 收稿日期:  2022-12-20
  • 修回日期:  2023-02-03
  • 上网日期:  2023-04-11
  • 刊出日期:  2023-04-20

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