Multi-objective and multi-constraint optimization of ultracold molecular orientation with limited rotational states

YU Zhenyang; HONG Qianqian; YI Yougen; SHU Chuancun

doi:10.7498/aps.74.20250684

Abstract

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 population 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.

Keywords:

Authors and contacts

Hunan Key Laboratory of Super-Microstructure and Ultrafast Process, School of Physics, Central South University, Changsha 410083, China

Corresponding author: SHU Chuancun, cc.shu@csu.edu.cn

Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12274470, 61973317) and the Natural Science Foundation of Hunan Province for Distinguished Young Scholars, China (Grant No. 2022JJ10070).

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References

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[20]	Qi D, Cao F, Xu S, et al. 2021 Phys. Rev. Appl. 15 024051 Google Scholar
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Cited By

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.

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图 2 无非目标态空间无约束少目标态模型优化结果: 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. 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.

DownLoad: Full-Size Img PowerPoint

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)和对应的转动态布居演化 (f)曲线

Figure 3. 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).

DownLoad: Full-Size Img PowerPoint

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)随迭代次数的变化

Figure 4. 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.

DownLoad: Full-Size Img PowerPoint

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. Time-dependent populations (a) and phases (b) for the model consisting of three rotational states in the target subspace and two rotational states in the non-target subspace.

DownLoad: Full-Size Img PowerPoint

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. 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.

DownLoad: Full-Size Img PowerPoint

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. 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.

DownLoad: Full-Size Img PowerPoint

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. 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.

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图 9 17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. 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.

DownLoad: Full-Size Img PowerPoint

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[2]	罗嗣佐, 陈洲, 李孝开, 胡湛, 丁大军 2019 光学学报 39 0126007 Google Scholar Luo S Z, Chen Z, Li X K, Hu Z, Ding D J 2019 Acta Opt. Sin. 39 0126007 Google Scholar
[3]	Lian Z, Luo S, Qi H, Chen Z, Shu C C, Hu Z 2023 Opt. Lett. 48 411 Google Scholar
[4]	Guo Y, Yang C, Xie X, Li Y, Houk K N, Guo X 2025 Sci. Adv. 11 eads0503 Google Scholar
[5]	Dong B, Pei Y, Mansour N, Lu X, Yang K, Huang W, Fang N 2019 Nat. Commun. 10 4815 Google Scholar
[6]	Cai M R, Ye C, Dong H, Li Y 2022 Phys. Rev. Lett. 129 103201 Google Scholar
[7]	Guo Y, Gong X, Ma S, Shu C C 2022 Phys. Rev. A 105 013102 Google Scholar
[8]	Liu Y, Meng J Q, Sun Z, Shu C C 2024 J. Phys. Chem. Lett. 15 8393 Google Scholar
[9]	Ploenes L, Straňák P, Mishra A, Liu X, Pérez-Ríos J, Willitsch S 2024 Nat. Chem. 16 1876 Google Scholar
[10]	Sawant R, Blackmore J A, Gregory P D, Mur-Petit J, Jaksch D, Aldegunde J, Hutson J M, Tarbutt M R, Cornish S L 2020 New J. Phys. 22 013027 Google Scholar
[11]	Ye J, Zoller P 2024 Phys. Rev. Lett. 132 190001 Google Scholar
[12]	Cornish S L, Tarbutt M R, Hazzard K R 2024 Nat. Phys. 20 730 Google Scholar
[13]	Ding M, Li J S, Deng J, Lee M C, Jolly J, Shahine B, Pawlicki T, Ma C M 2012 J. Chem. Phys. 137 265 Google Scholar
[14]	Tutunnikov I, Gershnabel E, Gold S, Averbukh I S 2018 J. Phys. Chem. Lett. 9 1105 Google Scholar
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[16]	DeMille D 2002 Phys. Rev. Lett. 88 067901 Google Scholar
[17]	Albert V V, Covey J P, Preskill J 2020 Phys. Rev. X 10 031050 Google Scholar
[18]	Nalewajski R F 2014 J. Math. Chem. 52 1292 Google Scholar
[19]	Pickering J D, Shepperson B, Hübschmann B A, Thorning F, Stapelfeldt H 2018 Phys. Rev. Lett. 120 113202 Google Scholar
[20]	Qi D, Cao F, Xu S, et al. 2021 Phys. Rev. Appl. 15 024051 Google Scholar
[21]	Loesch H, Remscheid A 1990 J. Chem. Phys. 93 4779 Google Scholar
[22]	Friedrich B, Herschbach D 1991 Z. Phys. D 18 153 Google Scholar
[23]	Lemeshko M, Krems R V, Doyle J M, Kais S 2013 Mol. Phys. 111 1648 Google Scholar
[24]	Koch C P, Lemeshko M, Sugny D 2019 Rev. Mod. Phys. 91 035005 Google Scholar
[25]	Nautiyal V V, Devi S, Tyagi A, Vidhani B, Maan A, Prasad V 2021 Spectrochim. Acta, Part A: Mol. Biomol. Spectrosc. 256 119663 Google Scholar
[26]	Hong Q Q, Lian Z Z, Shu C C, Henriksen N E 2023 Phys. Chem. Chem. Phys. 25 32763 Google Scholar
[27]	Dion C, Keller A, Atabek O 2001 Eur. Phys. J. D 14 249 Google Scholar
[28]	Machholm M, Henriksen N E 2001 Phys. Rev. Lett. 87 193001 Google Scholar
[29]	Babilotte P, Hamraoui K, Billard F, Hertz E, Lavorel B, Faucher O, Sugny D 2016 Phys. Rev. A 94 043403 Google Scholar
[30]	Shu C C, Yuan K J, Hu W H, Cong S L 2010 J. Chem. Phys. 132 244311 Google Scholar
[31]	Fleischer S, Zhou Y, Field R W, Nelson K A 2011 Phys. Rev. Lett. 107 163603 Google Scholar
[32]	Shu C C, Hong Q Q, Guo Y, Henriksen N E 2020 Phys. Rev. A 102 063124 Google Scholar
[33]	Tutunnikov I, Xu L, Field R W, Nelson K A, Prior Y, Averbukh I S 2021 Phys. Rev. Res. 3 013249 Google Scholar
[34]	De S, Znakovskaya I, Ray D, Anis F, Johnson N G, Bocharova I A, Magrakvelidze M, Esry B D, Cocke C L, Litvinyuk I V, Kling M F 2009 Phys. Rev. Lett. 103 153002 Google Scholar
[35]	Oda K, Hita M, Minemoto S, Sakai H 2010 Phys. Rev. Lett. 104 213901 Google Scholar
[36]	Znakovskaya I, Spanner M, De S, Li H, Ray D, Corkum P, Litvinyuk I V, Cocke C L, Kling M F 2014 Phys. Rev. Lett. 112 113005 Google Scholar
[37]	Ren X, Makhija V, Li H, Kling M F, Kumarappan V 2014 Phys. Rev. A 90 013419 Google Scholar
[38]	Lin K, Tutunnikov I, Qiang J, et al 2018 Nat. Commun. 9 5134 Google Scholar
[39]	Xu S, Lian Z, Hong Q Q, Wang L, Chen H, Huang Y, Shu C C 2024 Phys. Rev. A 110 023116 Google Scholar
[40]	Kitano K, Ishii N, Itatani J 2011 Phys. Rev. A 84 053408 Google Scholar
[41]	Shu C C, Henriksen N E 2013 Phys. Rev. A 87 013408 Google Scholar
[42]	Egodapitiya K N, Li S, Jones R R 2014 Phys. Rev. Lett. 112 103002 Google Scholar
[43]	Damari R, Kallush S, Fleischer S 2016 Phys. Rev. Lett. 117 103001 Google Scholar
[44]	Zhang S, Lu C, Jia T, Wang Z, Sun Z 2011 Phys. Rev. A 83 043410 Google Scholar
[45]	Yun H, Kim H T, Kim C M, Nam C H, Lee J 2011 Phys. Rev. A 84 065401 Google Scholar
[46]	Spanner M, Patchkovskii S, Frumker E, Corkum P 2012 Phys. Rev. Lett. 109 113001 Google Scholar
[47]	Qin C C, Jia G R, Zhang X Z, Liu Y F, Long J Y, Zhang B 2013 Chin. Phys. B 23 013302 Google Scholar
[48]	Huang Z Y, Wang D, Lang Z, Li W K, Zhao R R, Leng Y X 2017 Chin. Phys. B 26 054209 Google Scholar
[49]	Mun J H, Sakai H 2018 Phys. Rev. A 98 013404 Google Scholar
[50]	Li H, Li W, Feng Y, Pan H, Zeng H 2013 Phys. Rev. A 88 013424 Google Scholar
[51]	Cheng Q Y, Song Y Z, Meng Q T 2019 Chin. Phys. B 28 113301 Google Scholar
[52]	Damari R, Beer A, Flaxer E, Fleischer S 2023 J. Chem. Phys. 158 014201 Google Scholar
[53]	Kitano K, Ishii N, Kanda N, Matsumoto Y, Kanai T, Kuwata-Gonokami M, Itatani J 2013 Phys. Rev. A 88 061405 Google Scholar
[54]	Kitano K, Ishii N, Kanai T, Itatani J 2014 Phys. Rev. A 90 041402 Google Scholar
[55]	Zhang Y D, Wang S, Zhan W S, Yang J, Jing D 2017 Laser Phys. 27 056001 Google Scholar
[56]	Xu L, Tutunnikov I, Gershnabel E, Prior Y, Averbukh I S 2020 Phys. Rev. Lett. 125 013201 Google Scholar
[57]	Dion C M, Keller A, Atabek O 2005 Phys. Rev. A 72 023402 Google Scholar
[58]	Salomon J, Dion C M, Turinici G 2005 J. Chem. Phys. 123 144310 Google Scholar
[59]	Nakajima K, Abe H, Ohtsuki Y 2012 J. Phys. Chem. A 116 11219 Google Scholar
[60]	Liao S L, Ho T S, Rabitz H, Chu S I 2013 Phys. Rev. A 87 013429 Google Scholar
[61]	Coudert L H 2017 J. Chem. Phys. 146 024303 Google Scholar
[62]	Coudert L 2018 J. Chem. Phys. 148 094306 Google Scholar
[63]	Trippel S, Mullins T, Müller N L M, Kienitz J S, González-Férez R, Küpper J 2015 Phys. Rev. Lett. 114 103003 Google Scholar
[64]	Wang S, Henriksen N E 2020 Phys. Rev. A 102 063120 Google Scholar
[65]	Hong Q Q, Fan L B, Shu C C, Henriksen N E 2021 Phys. Rev. A 104 013108 Google Scholar
[66]	Fan L B, Shu C C, Dong D, He J, Henriksen N E, Nori F 2023 Phys. Rev. Lett. 130 043604 Google Scholar
[67]	Fan L B, Shu C C 2023 J. Phys. A-Math. Theor. 56 365302 Google Scholar
[68]	Zhang J P, Wang S, Henriksen N E 2023 Phys. Rev. A 107 033118 Google Scholar
[69]	Fan L B, Li H J, Chen Q, Zhou H, Liu H, Shu C C 2025 Phys. Rev. A 111 033119 Google Scholar
[70]	Hong Q Q, Dong D, Henriksen N E, Nori F, He J, Shu C C 2025 Phys. Rev. Res. 7 L012049 Google Scholar
[71]	Werschnik J, Gross E 2007 J. Phys. B: At. Mol. Opt. Phys. 40 R175 Google Scholar
[72]	Yoshida M, Ohtsuki Y 2014 Phys. Rev. A 90 013415 Google Scholar
[73]	Shu C C, Ho T S, Rabitz H 2016 Phys. Rev. A 93 053418 Google Scholar
[74]	Shu C C, Dong D, Petersen I R, Henriksen N E 2017 Phys. Rev. A 95 033809 Google Scholar
[75]	Guo Y, Dong D, Shu C C 2018 Phys. Chem. Chem. Phys. 20 9498 Google Scholar
[76]	Yu H, Ho T S, Rabitz H 2018 Phys. Chem. Chem. Phys. 20 13008 Google Scholar
[77]	Ansel Q, Dionis E, Arrouas F, Peaudecerf B, Guérin S, Guéry-Odelin D, Sugny D 2024 J. Phys. B: At. Mol. Opt. Phys. 57 133001 Google Scholar
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Vol. 74, Issue 18 - 2025-09-20

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