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利用数值模拟含时薛定谔方程的方法, 研究了氢原子在强激光场作用下产生的阈下高次谐波辐射. 在波长为608 nm激光作用下, 氢原子可以吸收5个光子, 由基态$\left|1s\right\rangle$ 共振跃迁至激发态$\left|2p\right\rangle$ 态; 氢原子也可以吸收更多光子发生电离, 某些连续态$\left|c\right\rangle$ 在激光缀饰下其能量可以和$\left|2p\right\rangle$ 态能量相当. 随后发生的$\left|2p\right\rangle$态到$\left|1s\right\rangle$ 态的复合, 或$\left|c\right\rangle$ 态到$\left|1s\right\rangle$ 态的复合, 可辐射出能量相同的光子. 这两个复合路径相互干涉, 在第五次谐波中形成典型的法诺线形. 进一步研究发现, 该法诺线形依赖于激光强度. 本研究表明, 即便是在单电子体系中, 法诺干涉也可以存在, 并可以通过控制激光参数改变法诺线形.We numerically solved the time-dependent Schrödinger equation (TDSE) for a hydrogen atom interacting with intense near-infrared laser fields to investigate the mechanism of below-threshold high-harmonic generation (HHG). The primary focus was on understanding the spectral features, particularly resonant structures, arising in the fifth harmonic region under specific driving conditions. Our simulations utilized a laser wavelength of 608 nm. At this wavelength, hydrogen atoms can resonantly absorb five photons, promoting electrons from the ground state $\left|1s\right\rangle$ to the excited state $\left|2p\right\rangle$. Concurrently, atoms can absorb additional photons leading to ionization. Crucially, due to the AC Stark shift induced by the intense laser field (laser dressing), certain laser-dressed continuum states $\left|c\right\rangle$ become energetically degenerate with the laser-dressed $\left|2p\right\rangle$ state. High-harmonic radiation at the fifth harmonic frequency can then be emitted via two distinct quantum paths: (1) Bound-bound recombination: Direct recombination from the laser-dressed $\left|2p\right\rangle$ state back to the ground state $\left|1s\right\rangle$. (2) Continuum-bound recombination: Recombination from the laser-dressed continuum states $\left|c\right\rangle$ (reached via ionization) back to $\left|1s\right\rangle$. Both pathways emit photons of identical energy corresponding to the fifth harmonic. Our key finding is the pronounced quantum interference between these two recombination channels. This interference manifests spectrally as a characteristic asymmetric Fano lineshape in the intensity profile of the fifth harmonic. Furthermore, we demonstrate that the shape of this Fano resonance exhibits a strong and controllable dependence on the intensity of the driving laser field. This study provides clear evidence that Fano quantum interference, typically associated with multi-electron correlations or autoionizing states in complex systems, can emerge in the fundamental single-electron hydrogen atom system under intense laser fields. The interference arises directly from the coherent superposition of the bound-bound and continuum-bound recombination pathways enabled by laser-induced degeneracy. Importantly, the spectral profile of the Fano resonance can be actively manipulated by tuning the laser intensity, highlighting a novel avenue for coherent control of harmonic emission in simple atomic systems.
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Keywords:
- Hydrogen atom /
- Intense laser field /
- Below-threshold high-harmonic generation /
- Fano resonance
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图 1 (a) 计算得到的高次谐波谱. 虚线标记了阈上和阈下谐波能量的分界点. (b) 为图(a)中第五次谐波的局部放大图. 激光波长为608 nm, 强度为5 × 1012 W cm–2
Fig. 1. (a) Calculated high-harmonic spectrum. The dashed line marks the boundary between above-threshold and below-threshold harmonic energies. (b) Partial enlarged view of the fifth harmonic in Fig. (a). The laser wavelength is 608 nm with an intensity of 5 × 1012 W cm–2.
图 2 双纵轴图: 随时间演化的态概率. 左轴为基态$ \left |1 s \right \rangle $, 右轴为激发态$ \left | 2 p \right \rangle $与连续态$ \left | c \right \rangle $的概率分布
Fig. 2. Dual-axis plot: Time evolution of state probabilities. The left axis corresponds to the ground state $ \left |1 s \right \rangle $, and the right axis shows the probabilities of the excited state $ \left | 2 p \right \rangle $ and continuum state $ \left | c \right \rangle $.
图 3 (a) 归一化的第五次谐波强度随激光强度变化的谱图, 激光强度从1012 W cm–2扫描至2 × 1013 W cm–2; (b) 对应三种典型强度(黑: 5 × 1012 W cm–2, 红: 1013 W cm–2, 蓝: 1.6 × 1013 W cm–2)下的谱线形; (c) 使用简化二能级模型所得结果, 仅包含$ \left | 1 s \right \rangle - $$ \left | 2 p \right \rangle $偶极路径
Fig. 3. (a) Spectral map of normalized fifth harmonic intensity versus laser intensity scanned from 1012 W cm–2 to 2 × 1013 W cm–2. (b) Spectral line shapes at three representative intensities (black: 5 × 1012 W cm–2, red: 1013 W cm–2, blue: 1.6 × 1013 W cm–2). (c) Results obtained using a simplified two-level model considering only the dipole $ \left | 1 s \right \rangle - \left | 2 p \right \rangle $ transition pathway.
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