Optimal control of population transfer in multi-level systems by dynamical quantum geometric tensor

LI Guanqiang; ZHANG Yuqi; GUO Hao; DONG Youjiao; LIN Zhiyu; PENG Ping

doi:10.7498/aps.74.20250210

Abstract

The optimal control of population transfer for multi-level systems is investigated from the perspective of quantum geometry. Firstly, the general theoretical framework of optimizing the STIRAP scheme based on the dynamical quantum geometric tensor is given, and then the dynamical quantum geometric tensor and the nonadiabatic transition rate are calculated by taking the detuned $ {{\Lambda }} $-type three-level system and tripod-type four-level system for example. Secondly, the transfer dynamics of the particle population of the system are investigated in detail. For a three-level system, the optimal STIRAP scheme has an efficiency of over 98% in transferring the population to the state $ \left|3\right.\rangle $, while the transfer efficiency of traditional STIRAP is about 72%. The superposition states with arbitrary proportions can be efficiently prepared for a four-level system due to the decoupling of the degenerate dark states. Finally, the influences of system parameters, such as the operating time of the Rabi pulses, the amplitude fluctuation and the single-photon detuning, on the transfer process are discussed. Especially, the phenomena of the adiabatic resonance transfer are revealed. Choosing the pulse parameters in the resonance window can reduce the infidelity of the population transfer to below 10^–3. It is found that the optimal STIRAP scheme by the dynamical quantum geometric tensor provides faster and more efficient transfer than the traditional STIRAP scheme.

Keywords:

Authors and contacts

School of Physics and Information Science, Shaanxi University of Science and Technology, Xi’an 710021, China

Corresponding author: LI Guanqiang, liguanqiang@sust.edu.cn

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References

[1]	Born M, Fock V 1928 Zeitschrift für Physik 51 165 Google Scholar
[2]	Wu Z, Yang H 2005 Phys. Rev. A 72 012114 Google Scholar
[3]	Holonomy S B 1983 Phys. Rev. Lett. 51 2167 Google Scholar
[4]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[5]	Vitanov N V, Rangelov A A, Shore B W, Bergmann K 2017 Rev. Mod. Phys. 89 015006 Google Scholar
[6]	Shore B W, Bergmann K, Oreg J, Rosenwaks S 1991 Phys. Rev. A 44 7442 Google Scholar
[7]	Shore B W 2013 Acta Phys. Slovaca 63 361 Google Scholar
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[13]	Fewell M P, Shore B W, Bergmann K 1997 Austra. J. Phys 50 281 Google Scholar
[14]	Guéry-Odelin D, Ruschhaupt A, Kiely A, Torrontegui E, Martínez-Garaot S, Muga J G 2019 Rev. Mod. Phys. 91 045001 Google Scholar
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[16]	Chen X, Lizuain I, Ruschhaupt A, Guéry-Odelin D, Muga J G 2010 Phys. Rev. Letts. 105 123003 Google Scholar
[17]	Minář Jí, Söyler Ş G, Rotondo P, Lesanovsky I 2017 New J. Phys. 19 063033 Google Scholar
[18]	Ban Y, Chen X, Sherman E Y, Muga J G 2012 Phys. Rev. Letts. 109 206602 Google Scholar
[19]	Opatrný T, Saberi H, Brion E, Mølmer K 2016 Phys. Rev. A 93 023815 Google Scholar
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[22]	Yu X M, Zhou K, Zhang H Y, Li S X, Huang Z, Wen J, Zhang R, Yu Y 2025 Phys. Rev. A 111 012623 Google Scholar
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[28]	Chen X, Torrontegui E, Muga J G 2011 Phys. Rev. A 83 062116 Google Scholar
[29]	Masuda S, Rice S A 2015 J. Phys. Chem. A 119 3479 Google Scholar
[30]	Masuda S, Nakamura K 2008 Phys. Rev. A 78 062108 Google Scholar
[31]	Torosov B T, Della Valle G, Longhi S 2014 Phys. Rev. A 89 063412 Google Scholar
[32]	Torosov B T, Della Valle G, Longhi S 2013 Phys. Rev. A 87 052502 Google Scholar
[33]	Li G Q, Chen G D, Peng P, Qi W 2017 Euro. Phys. J. D 71 14 Google Scholar
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[45]	Madasu C S, Rathod K D, Kwong C C, Wilkowski D 2024 Phys. Rev. Appl. 21 L051001 Google Scholar
[46]	Shi Z C, Wang J H, Zhang C, Song J, Xia Y 2024 Phys. Rev. A 109 022441 Google Scholar
[47]	Jin Z Y, Jing J 2025 Phys. Rev. A 111 022628 Google Scholar
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Cited By

图 1 具有单光子失谐的三能级系统STIRAP方案的示意图

Figure 1. Schematic diagram of the STIRAP scheme for a three-level system with single photon detuning

DownLoad: Full-Size Img PowerPoint

图 2 三能级系统拉比脉冲结构与粒子数布居的演化结果　(a)优化STIRAP脉冲结构; (b)优化STIRAP粒子数布居的演化结果; (c)传统STIRAP脉冲结构; (d)传统STIRAP粒子数布居的演化结果. 脉冲工作时间$ \tau =4{\mathrm{ }}\;{\text{μ}}{\mathrm{s}} $, 脉冲峰值$ {\varOmega }_{0}=30.79\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 失谐量$ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 2. Rabi pulse’s structures and the evolution results of the populations for the three-level system: (a) Pulse structures for the optimal STIRAP; (b) evolution of the populations for the optimal STIRAP; (c) pulse structures for the standard STIRAP; (d) evolution of the populations for the standard STIRAP. The pulse operating time $ \tau =4\;{\mathrm{ }}{\text{μ}}{\mathrm{s}} $, the pulse peak $ {\varOmega }_{0}=30.79{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, and the detuning $ \varDelta =2{\mathrm{\pi }}\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

DownLoad: Full-Size Img PowerPoint

图 3 混合角随时间的变化. 红色实线表示优化STIRAP的混合角, 蓝色虚线表示传统STIRAP的混合角

Figure 3. Change of the mixing angle with time. The red solid line indicates the mixing angle for the optimal STIRAP and the blue dashed line indicates the mixing angle for the standard STIRAP.

DownLoad: Full-Size Img PowerPoint

图 4 三能级系统失真度随工作时间的变化　(a)不存在失谐的情况($ \varDelta =0 $); (b)存在失谐的情况($ \varDelta =2{\mathrm{\pi }}\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}}) $. 红色实线表示优化STIRAP, 蓝色虚线表示传统STIRAP; 脉冲峰值$ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 4. Change of the infidelity with time for the three-level system: (a) The case without detuning ($ \varDelta =0) $; (b) the case with detuning ($ \varDelta =2{\mathrm{\pi }}{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). The red solid line corresponds to the optimal STIRAP scheme and the blue dashed line corresponds to the standard STIRAP one. The pulse peak $ {\varOmega }_{0}=35\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $.

DownLoad: Full-Size Img PowerPoint

图 5 三能级系统失真度随脉冲峰值涨落的变化　(a)不存在失谐的情况($ \varDelta =0 $); (b)存在失谐的情况($ \varDelta =2{\mathrm{\pi }}\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). 红色实线表示优化STIRAP, 蓝色虚线表示传统STIRAP; 脉冲峰值$ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 工作时间τ = 7.4 μs

Figure 5. Change of the infidelity with the fluctuation of the pulse peak for the three-level system: (a) The case without detuning ($ \varDelta =0 $); (b) the case with detuning ($ \varDelta = $$ 2{\mathrm{\pi }}\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). The red solid line denotes the optimal STIRAP scheme and the blue dashed line denotes the standard STIRAP one. The pulse peak $ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $ and the operating time τ = 7.4 μs.

DownLoad: Full-Size Img PowerPoint

图 6 三能级系统失真度随单光子失谐量的变化. 红色实线表示优化STIRAP, 蓝色虚线表示传统STIRAP; 脉冲峰值$ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 脉冲工作时间τ = 7.4 μs

Figure 6. Change of the infidelity with single-photon detuning for the three-level system. The red solid line denotes the optimal STIRAP scheme and the blue dashed line denotes the standard STIRAP one. The pulse peak $ {\varOmega }_{0}=35\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $ and the pulse operating time τ = 7.4 μs.

DownLoad: Full-Size Img PowerPoint

图 7 具有单光子失谐的四能级系统STIRAP方案的示意图

Figure 7. Schematic diagram of the STIRAP scheme for a four-level system with single photon detuning.

DownLoad: Full-Size Img PowerPoint

图 8 四能级系统拉比脉冲结构与粒子布居数的演化结果　(a)优化STIRAP脉冲结构; (b)优化STIRAP粒子布居数的演化结果; (c)传统STIRAP脉冲结构; (d)传统STIRAP粒子布居数的演化结果. 工作时间$ \tau =4{\mathrm{ }}\;{\text{μ}}{\mathrm{s}} $, 脉冲峰值$ {\varOmega }_{0}=35\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 失谐量$ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 8. Rabi pulse’s structures and the evolution results of the populations for the four-level system: (a) Pulse structures for the optimal STIRAP; (b) evolution of the populations for the optimal STIRAP; (c) pulse structures for the standard STIRAP; (d) evolution of the populations for the standard STIRAP. The pulse operating time $ \tau =4{\mathrm{ }}\;{\text{μ}}{\mathrm{s}} $, the pulse peak $ {\varOmega }_{0}=35\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, and the detuning $ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $.

DownLoad: Full-Size Img PowerPoint

图 9 四能级系统失真度随工作时间的变化情况　(a)不存在失谐的情况($ \varDelta =0 $); (b)存在失谐的情况($ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). 红色实线表示优化STIRAP, 蓝色虚线表示传统STIRAP; 脉冲峰值$ {\varOmega }_{0}=35\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 9. Change of the infidelity with time for the four-level system: (a) The case without detuning ($ \varDelta =0) $; (b) the case with detuning ($ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). The red solid line corresponds to the optimal STIRAP scheme and the blue dashed line corresponds to the standard STIRAP one. The pulse peak $ {\varOmega }_{0}=35\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $.

DownLoad: Full-Size Img PowerPoint

图 10 四能级系统失真度随脉冲峰值涨落的变化　(a)不存在失谐的情况($ \varDelta =0 $); (b)存在失谐的情况($ \varDelta =2{\mathrm{\pi }}\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). 红色实线表示优化STIRAP, 蓝色虚线表示传统STIRAP; 脉冲峰值$ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 脉冲工作时间τ = 7.4 μs

Figure 10. Change of the infidelity with the fluctuation of the pulse peak for the four-level system: (a) The case without detuning ($ \varDelta =0 $); (b) the case with detuning ($ \varDelta = $$ 2{\mathrm{\pi }}\;{\mathrm{ }}{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $). The red solid line denotes the optimal STIRAP scheme and the blue dashed line denotes the standard STIRAP one. The pulse peak $ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $ and the operating time τ = 7.4 μs

DownLoad: Full-Size Img PowerPoint

图 11 四能级系统失真度随失谐量$ {\mathrm{\varDelta }} $的变化. 红色实线为最优STIRAP, 蓝色虚线为传统STIRAP; 脉冲峰值$ {\varOmega }_{0}= $$ 35\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 脉冲工作时间τ = 7.4 μs

Figure 11. Change of the infidelity with single-photon detuning for the four-level system. The red solid line denotes the optimal STIRAP scheme and the blue dashed line denotes the standard STIRAP one. The pulse peak $ {\varOmega }_{0}=35\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $ and the pulse operating time τ = 7.4 μs.

DownLoad: Full-Size Img PowerPoint

图 12 大失谐($ \varDelta > {\varOmega }_{0} $)情况下四能级系统布居数演化结果　(a)优化STIRAP演化结果; (b)传统STIRAP演化结果. 蓝色虚线为$ \left|1\right.\rangle $态量子数布居, 绿色点线为$ \left|2\right.\rangle $态量子数布居, 红色虚线为$ \left|3\right.\rangle $态量子数布居, 黑色实线为$ |4\rangle $态量子数布居; 工作时间τ = 7.4 μs, 脉冲峰值$ {\varOmega }_{0}= $$ 22.13\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 失谐量$ \varDelta =58.1\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 12. Population’s evolution of the four-level system for the large detuning ($ \varDelta > {\varOmega }_{0} $): (a) The result for the optimal STIRAP scheme; (b) the result for the standard STIRAP scheme. The blue dashed line, the green dotted line, the red dashed line and the black solid line correspond to the populations in the states $ \left|1\right.\rangle $, $ \left|2\right.\rangle $, $ \left|3\right.\rangle $ and $ \left|4\right.\rangle $, respectively. The pulse operating time $ \tau =7.4{\mathrm{ }}\;{\text{μ}}{\mathrm{s}} $, the pulse peak $ {\varOmega }_{0}=22.13\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, and the detuning $ \varDelta = $$ 58.1\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $.

DownLoad: Full-Size Img PowerPoint

图 13 脉演化结果随脉冲参数$ \chi $变化情况. 蓝色虚线为$ \left|1\right.\rangle $态粒子数布居, 绿色点线为$ \left|2\right.\rangle $态粒子数布居, 红色虚线为$ \left|3\right.\rangle $态粒子数布居, 黑色实线为$ \left|4\right.\rangle $态粒子数布居; 工作时间τ = 7.4 μs, 脉冲峰值$ {\varOmega }_{0}=35\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 失谐量$ \varDelta =2 {\mathrm{\pi }}{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 13. Change of the populations for the four-level system with the parameter $ \chi $. The blue dashed line, the green dotted line, the red dashed line and the black solid line correspond to the populations in the states $ \left|1\right.\rangle $, $ \left|2\right.\rangle $, $ \left|3\right.\rangle $ and $ \left|4\right.\rangle $, respectively. The pulse operating time $ \tau =7.4\;{\mathrm{ }}{\text{μ}}{\mathrm{s}} $, the pulse peak $ {\varOmega }_{0}=35\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, and the detuning $ \varDelta =2{\mathrm{\pi }}{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $.

DownLoad: Full-Size Img PowerPoint

图 14 布居数演化结果　(a)脉冲参数$ \eta $= $ {\mathrm{\pi }}/4 $; (b)脉冲参数$ \eta $= $ {\mathrm{\pi }}/6 $. 蓝色虚线为$ \left|1\right.\rangle $态粒子数布居, 绿色点线为$ \left|2\right.\rangle $态粒子数布居, 红色虚线为$ \left|3\right.\rangle $态粒子数布居, 黑色实线为$ \left|4\right.\rangle $态粒子数布居; 工作时间τ = 7.4 $ {\text{μ}}{\mathrm{s}} $, 脉冲峰值$ {\varOmega }_{0}= $$ 35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, 失谐量$ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $

Figure 14. Evolution results of the populations: (a) Pulse parameter $ \eta $= $ {\mathrm{\pi }}/4 $; (b) pulse parameter $ \eta $= $ {\mathrm{\pi }}/6 $. The blue dashed line, the green dotted line, the red dashed line and the black solid line correspond to the populations in the states $ \left|1\right.\rangle $, $ \left|2\right.\rangle $, $ \left|3\right.\rangle $ and $ \left|4\right.\rangle $, respectively. The pulse operating time $ \tau =7.4\;{\mathrm{ }}{\text{μ}}{\mathrm{s}} $, the pulse peak $ {\varOmega }_{0}=35{\mathrm{ }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $, and the detuning $ \varDelta =2{\mathrm{\pi }}\;{\mathrm{M}}{\mathrm{H}}{\mathrm{z}} $.

DownLoad: Full-Size Img PowerPoint

[1]	Born M, Fock V 1928 Zeitschrift für Physik 51 165 Google Scholar
[2]	Wu Z, Yang H 2005 Phys. Rev. A 72 012114 Google Scholar
[3]	Holonomy S B 1983 Phys. Rev. Lett. 51 2167 Google Scholar
[4]	Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959 Google Scholar
[5]	Vitanov N V, Rangelov A A, Shore B W, Bergmann K 2017 Rev. Mod. Phys. 89 015006 Google Scholar
[6]	Shore B W, Bergmann K, Oreg J, Rosenwaks S 1991 Phys. Rev. A 44 7442 Google Scholar
[7]	Shore B W 2013 Acta Phys. Slovaca 63 361 Google Scholar
[8]	孙晓鹏, 冯志芳, 李卫东, 贾锁堂 2007 物理学报 56 5727 Google Scholar Sun X P, Feng Z F, Li W D, Jia S T 2007 Acta Phys. Sin. 56 5727 Google Scholar
[9]	孟少英, 吴炜, 刘彬 2009 物理学报 58 6902 Google Scholar Meng S Y, Wu W, Liu B 2009 Acta Phys. Sin. 58 6902 Google Scholar
[10]	李冠强, 彭娉 2011 物理学报 60 110304 Google Scholar Li G Q, Peng P 2011 Acta Phys. Sin. 60 110304 Google Scholar
[11]	李冠强, 彭娉, 曹振洲, 薛具奎 2012 物理学报 61 090301 Google Scholar Li G Q, Peng P, Cao Z Z, Xue J K 2012 Acta Phys. Sin. 61 090301 Google Scholar
[12]	Bergmann K, Vitanov N V, Shore B W 2015 J. Chem. Phys. 142 170901 Google Scholar
[13]	Fewell M P, Shore B W, Bergmann K 1997 Austra. J. Phys 50 281 Google Scholar
[14]	Guéry-Odelin D, Ruschhaupt A, Kiely A, Torrontegui E, Martínez-Garaot S, Muga J G 2019 Rev. Mod. Phys. 91 045001 Google Scholar
[15]	Hatomura T 2024 J. Phys. B: At. Mol. Opt. Phys. 57 102001 Google Scholar
[16]	Chen X, Lizuain I, Ruschhaupt A, Guéry-Odelin D, Muga J G 2010 Phys. Rev. Letts. 105 123003 Google Scholar
[17]	Minář Jí, Söyler Ş G, Rotondo P, Lesanovsky I 2017 New J. Phys. 19 063033 Google Scholar
[18]	Ban Y, Chen X, Sherman E Y, Muga J G 2012 Phys. Rev. Letts. 109 206602 Google Scholar
[19]	Opatrný T, Saberi H, Brion E, Mølmer K 2016 Phys. Rev. A 93 023815 Google Scholar
[20]	Tian L 2012 Phys. Rev. Lett. 108 153604 Google Scholar
[21]	Barrett S, Hammerer K, Harrison S, Northup T E, Osborne T J 2013 Phys. Rev. Lett. 110 090501 Google Scholar
[22]	Yu X M, Zhou K, Zhang H Y, Li S X, Huang Z, Wen J, Zhang R, Yu Y 2025 Phys. Rev. A 111 012623 Google Scholar
[23]	Masuda S 2012 Phys. Rev. A 86 063624 Google Scholar
[24]	Masuda S, Güngördü U, Chen X, Ohmi T, Nakahara M 2016 Phys. Rev. A 93 013626 Google Scholar
[25]	Demirplak M, Rice S A 2003 J. Phys. Chem. A 107 9937 Google Scholar
[26]	Berry M V 2009 J. Phys. A: Math. Theor. 42 365303 Google Scholar
[27]	Lewis H R, Riesenfeld W B 1969 J. Math. Phys. 10 1458 Google Scholar
[28]	Chen X, Torrontegui E, Muga J G 2011 Phys. Rev. A 83 062116 Google Scholar
[29]	Masuda S, Rice S A 2015 J. Phys. Chem. A 119 3479 Google Scholar
[30]	Masuda S, Nakamura K 2008 Phys. Rev. A 78 062108 Google Scholar
[31]	Torosov B T, Della Valle G, Longhi S 2014 Phys. Rev. A 89 063412 Google Scholar
[32]	Torosov B T, Della Valle G, Longhi S 2013 Phys. Rev. A 87 052502 Google Scholar
[33]	Li G Q, Chen G D, Peng P, Qi W 2017 Euro. Phys. J. D 71 14 Google Scholar
[34]	Li K Z, Tian J Z, Xiao L T 2024 Phys. Rev. A 109 022443 Google Scholar
[35]	Chen J F 2022 Phys. Rev. Res. 4 023252 Google Scholar
[36]	Sun C P 1988 J. Phys. A: Math. Gen. 21 1595 Google Scholar
[37]	Rigolin G, Ortiz G, Ponce V H 2008 Phys. Rev. A 78 052508 Google Scholar
[38]	Chen J F, Sun C P, Dong H 2019 Phys. Rev. E 100 062140 Google Scholar
[39]	Oh S, Shim Y P, Fei J, et al. 2013 Phys. Rev. A 87 022332 Google Scholar
[40]	Gaubatz U, Rudecki P, Schiemann S, Bergmann K 1990 J. Chem. Phys. 92 5363 Google Scholar
[41]	Unanyan R G, Shore B W, Bergmann K 2001 Phys. Rev. A 63 043401 Google Scholar
[42]	Vitanov N V 1998 Phys. Rev. A 58 2295 Google Scholar
[43]	Vitanov N V, Halfmann T, Shore B W, Bergmann K 2001 Ann. Rev. Phys. Chem. 52 763 Google Scholar
[44]	Unanyan R, Fleischhauer M, Shore B W, Bergmann K 1998 Opt. Commun. 155 144 Google Scholar
[45]	Madasu C S, Rathod K D, Kwong C C, Wilkowski D 2024 Phys. Rev. Appl. 21 L051001 Google Scholar
[46]	Shi Z C, Wang J H, Zhang C, Song J, Xia Y 2024 Phys. Rev. A 109 022441 Google Scholar
[47]	Jin Z Y, Jing J 2025 Phys. Rev. A 111 022628 Google Scholar
[48]	Li G Q, Guo H, Zhang Y Q, Yang B, Peng P 2025 Commun. Theor. Phys. 77 015103 Google Scholar

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Vol. 74, Issue 10 - 2025-05-20

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