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在强场激发的一般情况下, 斯塔克效应对于瞬态双光子跃迁具有重要影响, 且该过程的解析描述具有很大挑战. 本文采用解析求解与数值模拟相结合的方法, 系统研究了弱场和强场啁啾脉冲激发的瞬态双光子跃迁过程, 揭示了光场强度、啁啾因子、失谐量等参数对双光子跃迁概率时域演化的重要影响. 首先, 本文利用二阶微扰理论得到了双光子时域跃迁概率振幅的近似解析解表达式. 该解析解表明, 弱场激发的瞬态双光子跃迁过程类似于菲涅耳直边衍射效应. 随着光场强度的增强, 斯塔克效应对双光子跃迁的影响随之增强. 其次, 本文通过一系列近似处理得到了强场作用下薛定谔方程的近似解析解. 此解析解表明, 强场斯塔克效应引起能级分裂使得双光子跃迁概率时域的对称性遭到了破坏, 其频域过程类似于“双缝干涉”效应. 研究结果表明, 强场激发时布居转移效率与光场强度具有重要关系, 而啁啾因子不仅可以调节布居转移效率和时间位置, 还可以改变布居概率在时域的振荡频率. 这对于强场激发的布居概率时域演化描述提供了新思路, 并对双光子显微成像研究提供了科学依据.In general cases of strong field excitation, the Stark effect has a significant influence on transient two-photon transitions, and the analytic description of this process is quite challenging. By combining analytical solutions and numerical simulations, the transient two-photon transition processes excited by weak and strong chirped pulses are systematically investigated, showing the important influences of parameters such as light field intensity, chirp factor, and detuning on the time-domain evolution of two-photon transition probabilities. Firstly, an approximate analytical expression is derived for the amplitude of the time-domain two-photon transition probability by using the second-order perturbation theory. This analytical solution indicates that the transient two-photon transition process under weak field excitation is similar to the Fresnel rectangular edge diffraction effect. As the light field intensity increases, the influence of the Stark effect on two-photon transitions also intensifies. Secondly, through a series of approximations, the approximate analytical solutions of the Schrödinger equation under strong field interactions are obtained. The analytical solutions show that the strong field Stark effect induces energy level to split, which disrupts the symmetry of the time-domain two-photon transition probability distribution, and its frequency domain process is similar to the “double-slit interference” effect. The research results indicate that the efficiency of population transfer during strong field excitation is closely related to the light field intensity, while the chirp factor can not only regulate the efficiency and time position of population transfer but also change the oscillation frequency of the population probability in the time domain. This work offers new insights into describing the time-domain evolution of the population probability under strong field excitation and lays a scientific basis for research on two-photon microscopy imaging.
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Keywords:
- two-photon transition /
- transient process /
- perturbation theory /
- chirp factor
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图 1 飞秒啁啾脉冲驱动下双光子跃迁的激发方案 (a) 二能级原子系统中双光子跃迁模型; (b) 飞秒啁啾脉冲的高斯线形包络(蓝色)和相位分布(橙色)
Fig. 1. Excitation scheme of two-photon transition by a femtosecond chirped pulse: (a) Two-photon transition model in a two-level atomic system; (b) Gaussian profile envelope (blue) and the phase distribution (orange) of the femtosecond chirped pulse.
图 2 弱场激发时终态布居概率以及波函数实部和虚部随着时间和啁啾因子的变化 (a) 终态布居概率随着时间和啁啾因子的变化; (b), (c) 图(a)中两个不同啁啾因子值(红色α = –0.05 fs2和蓝色圆圈α = 0)条件下终态布居概率随着时间的演化; (d) 终态波函数实部随着时间和啁啾因子的变化; (e), (f) 图(d)中两个不同啁啾因子值(红色和蓝色圆圈)条件下终态波函数实部随着时间的演化; (g) 终态波函数虚部随着时间和啁啾因子的变化; (h), (i) 图(g)中两个不同啁啾因子值(红色和蓝色圆圈)条件下终态波函数虚部随着时间的演化
Fig. 2. Evolution of the population probability of the final-state and the real-imaginary part of the wave-function with time and chirp factor under a weak field excitation: (a) The population probability of the final-state versus time and detuning; (b), (c) the population probability at two different chirp factors (red and blue circles) in panel (a); (d) the real part of the wave-function of the final-state versus time and the chirp factor; (e), (f) the real part of the wave-function at two different chirp factors (red and blue circles) in panel (d); (g) the imaginary part of the wave-function of the final-state versus time and the chirp factor; (h), (i) the imaginary part of the wave-function at two different chirp factors (red and blue circles) in panel (g).
图 3 弱场作用下终态布居概率及波函数实部和虚部随着时间和失谐量的变化 (a) 终态布居概率随着时间和失谐量的变化; (b) 图(a)中3个不同失谐量值(–1500 THz, –600 THz, 200 THz)条件下终态布居概率随着时间的演化; (c) 终态波函数实部随着时间和失谐量的变化; (d) 图(c)中3个不同失谐量值(–1500 THz, –600 THz, 200 THz)条件下终态波函数实部随着时间的变化; (e) 终态波函数虚部随着时间和失谐量的变化; (f) 图(e)中3个不同失谐量值(–1500 THz, –600 THz, 200 THz)条件下终态波函数虚部随着时间的变化
Fig. 3. Evolution of the population probability and the real-imaginary part of the final-state with time and detuning under a weak field excitation: (a) The population probability of the final-state versus time and detuning; (b) the population probability of the final-state versus time at three different detunings (–1500 THz, –600 THz, 200 THz) in panel (a); (c) the real part of the wave-function of the final-state versus time and the detuning; (d) the real part of the wave-function of the final-state versus time at three different detunings (–1500 THz, –600 THz, 200 THz) in panel (c); (e) the imaginary part of the wave-function of the final-state versus time and the detuning; (f) the imaginary part of the wave-function of the final-state versus time at three different detunings (–1500 THz, –600 THz, 200 THz) in panel (e).
图 5 强场作用下基态与终态布居概率随着时间的变化(a)—(f)分别对应不同啁啾因子情况; (g)—(l)分别对应不同光强情况
Fig. 5. Evolution of the population probability of the ground-state and final-state with time under a strong field excitation: (a)–(f)Under different chirp factor; (g)–(l) under different intensity of laser field.
图 6 强场作用下终态布居概率随着时间和失谐量的变化 (a)—(c) 正啁啾因子情况; (d)—(f) 负啁啾因子情况
Fig. 6. Evolution of the population probability of the final-state with time and detuning under a strong field excitation: (a)–(c) With positive chirp factors; (d)–(f) with negative chirp factors.
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