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The design of shaping pulse fields for controlling molecular orientation is of great importance in fields of stereochemical reactions, strong-field ionization, and quantum information processing. Traditional quantum optimal control algorithms typically solve the problem of molecular orientation in an infinite-dimensional rotational space, but they often overlook the constraints imposed by experimental limitations. In this work, a multi-objective and multi-constraint quantum optimal control algorithm is proposed to design a pulse field that conforms to the constraints of pulse area and energy. Specifically, the algorithm enforces a zero pulse area condition to eliminate the static field components and maintains constant pulse energy, ensuring compatibility with realistic experimental setups. Under these constraints, the algorithm optimizes the number and phase distribution of a selected number of low-lying rotational states in ultracold molecules to achieve maximum molecular orientation. The effectiveness of the proposed algorithm is demonstrated through numerical studies involving two- and three-state target subspaces, where the creation of a coherent superposition state with optimized population and phase distribution leads to the desired molecular orientation. Furthermore, its scalability is validated by applying it to a more complex 17-state subspace, where a maximum orientation value of 0.99055 is obtained, approaching the global optimal value of 1. Our findings demonstrate that by effectively managing these constraints, the influence of rotational states in the non-target state subspace can be substantially suppressed. The time-frequency analysis of the optimized pulses, combined with the Fourier transform spectrum of the time-dependent degree of orientation, indicates that the maximum molecular orientation is mainly achieved through ladder-climbing excitation of multi-color pulse fields, with the contributions from highly excited states being minimal. This work provides a valuable reference for designing experimentally feasible pulse fields using multi-constraint optimization algorithms, which helps to precisely control a limited number of rotational states to achieve maximum molecular orientation.
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Keywords:
- Quantum optimal control /
- Molecular orientation /
- Multiple constraints /
- Multiple objectives
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图 1 脉冲场激发分子转动态调控分子取向模型示意图. 下面的蓝色能级表示目标态空间含有M个最低转动态; 灰色能级表示非目标态空间包含N个转动态. 本文通过使用多目标多约束量子最优控制理论算法寻找优化脉冲场调控目标空间转动态分布, 从而获得最优分子取向
Figure 1. Schematic diagram of the molecular rotation states excited by the pulse field. The blue lines indicate that the target subspace consists of the lowest M rotational states, while the gray lines denote that the non-target subspace contains N rotational states above the target.
图 2 无非目标态空间无约束少目标态态模型优化结果: 上排表示仅两态模型($ M = 2 $, $ N = 0 $)最大取向度 (a)、脉冲面积 (b)和脉冲能量 (c)随迭代次数变化曲线; 下排表示将优化获得脉冲 (d)用于四态模型($ M = 2 $, $ N = 2 $)所得的含时取向度 (e)及对应的转动态布居演化 (f)曲线
Figure 2. The non-constrained optimization results for the model comprising two rotational states within the target subspace. The upper panels depict the maximum degree of orientation (a), pulse area (b), and pulse energy (c) as a function of interactions by excluding the non-target subspace. The lower panels illustrate the optimized time-dependent control fields (d) for the four-state model ($ M = 2 $, $ N = 2 $), alongside the corresponding orientation (e) and the population evolution (f) of the rotational states.
图 3 含非目标态空间和约束条件两目标态模型($ M = 2 $, $ N = 2 $)优化结果: 上排最大取向度 (a)、脉冲面积 (b)和脉冲能量 (c)随迭代次数变化曲线; 下排表示优化获得脉冲 (d)及含时取向度 (e)及对应的转动态布居演化 (f)曲线
Figure 3. The constrained optimization results for the model comprising two rotational states in the target subspace and two rotational state in the non-target subspace. The upper panels depict the maximum degree of orientation (a), pulse area (b), and pulse energy (c) as a function of interactions. The lower panels illustrate the optimized time-dependent control fields (d), alongside the corresponding orientation (e) and the population evolution of the rotational states (f).
图 4 含非目标态空间和约束条件少目标态模型($ M = 2 $, $ N = 2 $)优化结果: 最大取向度 (a)、非目标态空间总布居 (b), 脉冲面积 (c)和脉冲能量 (d)随迭代次数变化曲线
Figure 4. The constrained optimization results for the model comprising three rotational states in the target subspace and two rotational state in the non-target subspace. The maximum orientation (a), the total population in the non-target subspace (b), the pulse area (c), and the pulse energy (d) as a function of iterations.
图 5 三目标态模型($ M = 2 $, $ N = 2 $)优化转动态布居 (a)和相位 (b)演化曲线
Figure 5. The time-dependent populations and phases for the model consisting of three rotational states in the target subspace and two rotational states in the non-target subspace.
图 6 三目标态模型($ M = 2 $, $ N = 2 $)优化含时脉冲场 (a), 对应的时频谱 (b), 含时取向度 (c)以及对应傅里叶变换谱 (d)
Figure 6. The optimal time-dependent pulse field (a), the corresponding time- and frequency-resolved distributions (b), the time-dependent orientation degree (c), and the corresponding Fourier transform spectrum (d) for the model consisting of three rotational states in the target subspace and two rotational states in the non-target subspace.
图 7 含非目标态空间和约束条件多目标态模型($ M = 17 $, $ N = 2 $)优化结果: 最大取向度 (a)、非目标态空间总布居 (b), 脉冲面积 (c)和脉冲能量 (d)随迭代次数变化曲线
Figure 7. The constrained optimization results for the model comprising seventeen rotational states in the target subspace and two rotational states in the non-target subspace. The maximum orientation (a), the total population in the non-target subspace (b), the pulse area (c), and the pulse energy (d) as a function of iterations.
图 8 优化的17态布居 (a)及对应的相位 (b)分布, (a)中红色三条符号代表理论计算的最优布居分布
Figure 8. The optimal population (a) and phase (b) distribution of 17 rotational states in the target subspace. The red triangles in (a) denote the analyzed population distributions.
图 9 十七目标态模型($ M = 17 $, $ N = 2 $)优化含时脉冲场 (a), 对应的时频谱 (b), 含时取向度 (c)以及对应傅里叶变换谱 (d)
Figure 9. The optimal time-dependent pulse field (a), the corresponding time- and frequency-resolved distributions (b), the time-dependent orientation degree (c), and the corresponding Fourier transform spectrum (d) for the model consisting of seventeen rotational states in the target subspace and two rotational states in the non-target subspace.
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