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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)
Figure 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
Figure 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
Figure 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
Figure 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
Figure 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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