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自旋-轨道耦合玻色爱因斯坦凝聚中多能级绝热消除理论

袁家望 陈立 张云波

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自旋-轨道耦合玻色爱因斯坦凝聚中多能级绝热消除理论

袁家望, 陈立, 张云波

Adiabatic elimination theory of multi-level system in spin-orbit coupled Bose-Einstein condensate

Yuan Jia-Wang, Chen Li, Zhang Yun-Bo
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  • 在量子光学中, 绝热消除可以简化多能级量子系统, 它通过消除快速振荡自由度、保留慢变量动力学, 从而获得系统的有效描述. 绝热消除在量子模拟和量子精密测量中具有重要应用, 比如利用三能级拉曼耦合和绝热消除人们在超冷原子中实现了自旋-轨道耦合. 本文在三能级绝热消除的基础上研究三能级非厄米系统与多能级系统中绝热消除的理论方法及其推广, 验证了绝热消除理论在非厄米系统和多能级系统中的有效性与准确性. 本文研究可为耗散的多能级量子系统中的态制备和动力学操控提供理论基础.
    In quantum optics, adiabatic elimination simplifies multi-level quantum system by eliminating the fast oscillatory degree of freedom and preserving the slow-varying dynamics, thus obtaining an efficient description of the system. Adiabatic elimination has important applications in quantum simulation and quantum precision measurement. For example, spin-orbit coupling has been realized in ultracold atoms by using three-level Raman coupling and adiabatic elimination. In this paper, we investigate the theoretical method and generalize the adiabatic elimination in three-level non-Hermitian systems and multi-level systems on the basis of standard elimination scheme. These can provide theoretical guidance for realizing the interdiscipline of non-Hermitian physics and spin-orbit coupling effects and their potential applications. We mainly discuss the influences of dissipative effect on the population dynamics of the system, the validity and accuracy of the adiabatic elimination theory under different parameters for both non-Hermitian and two types of five-level systems. Specifically, the dynamics satisfying the large detuning condition gives very accurate results for quite a long evolution time with the adiabatic elimination theory, but when the two-photon detuning δ and the Rabi frequency $\varOmega $ gradually increase, leading to the violation of the large detuning condition $ \varOmega,\gamma, \delta \ll \Delta$, the effective two-level model can no longer describe the fast-varying dynamics of the system even in a short evolution time. Thus the choice of system parameters affects the effectiveness of adiabatic elimination of the excited levels. In a non-Hermitian system, the population in the ground state oscillates with gain periodically at the beginning, while that in the ground state oscillates with loss and decreases with time, with the total population decreasing with oscillation. For long-time evolution the gain in the system causes the population to diverge, and the adiabatic elimination of the effective two-energy level system describes this behavior accurately. The effect of the non-Hermitian parameters on the dynamics of the system in the resonance case is manifested in the case that the total population remains conserved, while the total population tends to diverge for finite two-photon detuning. We find that with the increase of detuning, the divergence appears earlier and the total number of particles can be kept constant by choosing the ratio of gain to loss appropriately. This study provides a theoretical basis for state preparation and dynamical manipulation in dissipative multi-energy quantum systems.
      通信作者: 张云波, ybzhang@zstu.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 12074340, 12174236)和浙江理工大学科学基金(批准号: 20062098-Y)资助的课题.
      Corresponding author: Zhang Yun-Bo, ybzhang@zstu.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12074340, 12174236) and the Zhejiang Sci-Tech University of Technology Science Foundation, China (Grant No. 20062098-Y).
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    Kittel C 1963 Quantum Theory of Solids (New York: John Wiley and Sons

    [2]

    Goldman N, Juzeliūnas G, Öhberg P, Spielman I B 2014 Rep. Prog. Phys. 77 126401Google Scholar

    [3]

    Zheng W, Yu Z Q, Cui X L, Zhai H 2013 J. Phys. B 46 134007Google Scholar

    [4]

    Jian C M, Zhai H 2011 Phys. Rev. B. 84 060508Google Scholar

    [5]

    Zhang J Y, Ji S C, Chen Z, Zhang L, Du Z D, Yan B, Pan G S, Zhao B, Deng Y J, Zhai H, Chen S, Pan J W 2012 Phys. Rev. Lett. 109 115301Google Scholar

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    焦宸, 简粤, 张爱霞, 薛具奎 2023 物理学报 72 060302Google Scholar

    Jiao C, Jian Y, Zhang A X, Xue J K 2023 Acta Phys. Sin. 72 060302Google Scholar

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    施婷婷, 汪六九, 王璟琨, 张威 2020 物理学报 69 016701Google Scholar

    Shi T T, Wang L J, Wang J K, Zhang W 2020 Acta Phys. Sin. 69 016701Google Scholar

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    陈星, 薛潇博, 张升康, 马余全, 费鹏, 姜元, 葛军 2021 物理学报 70 083401Google Scholar

    Chen X, Xue X B, Zhang S K, Ma Y Q, Fei P, Jiang Y, Ge J 2021 Acta Phys. Sin. 70 083401Google Scholar

    [9]

    Campbell D L, Price R M, Putra A, Valdés-Curiel A, Trypogeorgos D, Spielman I B 2016 Nat. Commun. 7 10897Google Scholar

    [10]

    Galitski V, Spielman I B 2013 Nature. 494 49Google Scholar

    [11]

    Stanescu T D, Anderson B, Galitski V 2008 Phys. Rev. A. 78 023616Google Scholar

    [12]

    Xu P, Deng T S, Zheng W, Zhai H 2015 Phys. Rev. A 103 L061302Google Scholar

    [13]

    Wang P J, Yu Z Q, Fu Z k, Miao J, Huang L H, Chai S J, Zhai H, Zhang J 2012 Phys. Rev. Lett. 109 095301Google Scholar

    [14]

    Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S, Zwierlein M W 2012 Phys. Rev. Lett. 109 095302Google Scholar

    [15]

    Chen X L, Wang J, Li Y, Liu X J, Hu H 2018 Phys. Rev. A. 98 013614Google Scholar

    [16]

    Lan Z H, Öhberg P 2014 Phys. Rev. A. 89 023630Google Scholar

    [17]

    Chen L, Pu H, Zhang Y B 2016 Phys. Rev. A. 93 013629Google Scholar

    [18]

    Zhai H 2012 Int. J. Mod. Phys. B 26 1230001Google Scholar

    [19]

    Zhai H 2015 Rep. Prog. Phys. 78 026001Google Scholar

    [20]

    贺丽, 余增强 2017 物理学报 66 220301Google Scholar

    He L, Yu Z Q 2017 Acta Phys. Sin. 66 220301Google Scholar

    [21]

    Geier K T, Martone G I, Hauke P, Stringari S 2021 Phys. Rev. Lett. 127 115301Google Scholar

    [22]

    Liao R Y 2018 Phys. Rev. Lett. 120 140403Google Scholar

    [23]

    Chen L, Zhang Y B, Pu H 2020 Phys. Rev. Lett. 125 195303Google Scholar

    [24]

    Li Y, Pitaevskii L P, Stringari S 2012 Phys. Rev. Lett. 108 225301Google Scholar

    [25]

    Lin Y J, Jiménez-García K, Spielman I B 2011 Nature. 471 83Google Scholar

    [26]

    Steck D A 2007 Quantum and Atom Optics (Eugene, Oregon: Department of Physics, University of Oregon

    [27]

    Dalibard J, Gerbier F, Juzeliūnas G, Öhberg P 2011 Rev. Mod. Phys. 83 1523Google Scholar

    [28]

    Bergmann K, Theuer H, Shore B W 1998 Rev. Mod. Phys. 70 1003Google Scholar

    [29]

    Král P, Thanopulos I, Shapiro M 2007 Rev. Mod. Phys. 79 53Google Scholar

    [30]

    Brion E, Pedersen L H, Mølmer K 2007 J. Phys. A 40 1033Google Scholar

    [31]

    Li H, Shen H Z, Wu S L, Yi X X 2017 Optics Express. 25 30135Google Scholar

    [32]

    Li G Q, Chen G D, Peng P, Qi W 2017 Eur. Phys. J. D 71 14Google Scholar

    [33]

    Guo L P, Du L, Yin C H, Zhang Y B, Chen S 2018 Phys. Rev. A. 97 032109Google Scholar

  • 图 1  三能级Λ型系统能级图

    Fig. 1.  Three-level Λ type system

    图 2  三能级Λ系统布居数动力学. 单光子失谐$ \varDelta_1 = $$ 595\pi $ MHz, $ \varDelta_2 = 605\pi $ MHz及$ \varOmega_1 = \varOmega_2 = 120\pi $ MHz, 初态$ |\psi_0\rangle = |g_1\rangle $, 其中实线为消除前三能级布居数(绿线、蓝线、橙线分别代表$ |g_1\rangle $, $ |g_2\rangle $, $ |e\rangle $态), 圆点为消除后有效二能级系统布居数(绿点和蓝点分别代表$ |g_1\rangle $$ |g_2\rangle $态)

    Fig. 2.  Population dynamics of the three-level Λ system and the effective two-level system for $ \varDelta_1 = 595\pi $ MHz, $ \varDelta_2 = $$ 605\pi $ MHz and $ \varOmega_1 = \varOmega_2 = 120\pi $ MHz starting from the initial state $ |\psi_0\rangle = |g_1\rangle $. Green, blue, and orange line denote the population of $|g_1\rangle $, $ |g_2\rangle$, $|e\rangle $ state before adiabatic elimination, respectively. Green and blue dots denote the population of $|g_1\rangle $ and $ |g_2\rangle$ state after adiabatic elimination, respectively.

    图 3  三能级Λ系统$|g_1\rangle$态布居数动力学. 初态$|\psi_0\rangle= $$ |g_1\rangle$, 其中红线为消除前$|g_1\rangle$态布居数, 蓝线为消除后$|g_1\rangle$态布居数. 图(a)—(c)中, 双光子失谐和Rabi频率分别为$\delta=10\pi$ MHz, $\varOmega=60\pi$ MHz; $\delta=20\pi$ MHz, $\varOmega= $$ 120\pi$ MHz; $\delta=100\pi$ MHz, $\varOmega=240\pi$ MHz

    Fig. 3.  Population dynamics of the $|g_1\rangle$ state of the three-level Λ system for (a) $\delta = 10\pi$ MHz, $\varOmega = 60\pi$ MHz; (b) $\delta= $$ 20\pi$ MHz, $\varOmega=120\pi$ MHz; (c)$\delta=100\pi$ MHz, $\varOmega=240\pi$ MHz starting from the initial state $|\psi_0\rangle=|g_1\rangle$. Red and blue line denote the population of $|g_1\rangle $ state before and after adiabatic elimination, respectively.

    图 4  三能级非厄米Λ系统布居数动力学. 单光子失谐$\varDelta_1= $$ 595\pi$ MHz, $\varDelta_2=605\pi$ MHz以及$\varOmega_1=\varOmega_2=120\pi$ MHz, 耗散$\gamma_1=\gamma_2=0.2\pi$ MHz, 初态$|\psi_0\rangle=(|g_2\rangle-|g_1\rangle)/\sqrt{2}$. 其中实线为消除前三能级布居数(绿线、蓝线、橙线分别代表$|g_1\rangle$, $|g_2\rangle$, $|e\rangle$态), 圆点为消除后有效二能级系统布居数(绿点和蓝点分别代表$|g_1\rangle$$|g_2\rangle$态). 灰色虚线代表系统总布居数, 从下到上依次为双光子失谐$\delta=10\pi, 20\pi, 30\pi$ MHz. 特别地当双光子失谐为0时, 总布居数不会发散(黑色虚线)

    Fig. 4.  Population dynamics of the three-level Λ type non-Hermitian system for $\varDelta_1=595\pi$ MHz, $\varDelta_2=605\pi$ MHz and $\varOmega_1=\varOmega_2=120\pi$ MHz, with the dissipative parameter $\gamma_1=\gamma_2=0.2\pi$ MHz, starting from the initial state $|\psi_0\rangle=(|g_2\rangle-|g_1\rangle)/\sqrt{2}$. Green, blue, and orange line denote the population of $|g_1\rangle $, $ |g_2\rangle$, $|e\rangle $ state before adiabatic elimination, respectively. Green and blue dots denote the population of $|g_1\rangle $ and $ |g_2\rangle$ state after adiabatic elimination, respectively. The gray dashed line represents the total population of atoms, with two-photon detunings of 10π, 20π, 30π MHz from bottom to top. Specifically when the two photon detuning is 0, the total population will not diverge (see the black dashed line).

    图 5  双Λ型五能级系统能级图

    Fig. 5.  Five-level double-Λ type system

    图 6  双Λ型五能级系统数值解布居数动力学. 单光子失谐$\varDelta_1=590\pi$ MHz, $\varDelta_2=600\pi$ MHz, $\varDelta_3=600\pi$ MHz, $\varDelta_4=610\pi$ MHz, 拉比频率$\varOmega_1=\varOmega_2=\varOmega_3=\varOmega_4=120\pi$ MHz, 处在初态$|\psi_0\rangle=|-1\rangle$. 其中实线为消除前能级布居数(蓝线、绿线、橙线、紫线、灰线分别代表$|1\rangle$, $|0\rangle$, $|-1\rangle$, $|a\rangle$, $|b\rangle$态), 圆点为消除后有效三能级布居数(蓝点、绿点和橙点分别代表$|1\rangle$, $|0\rangle$, $|-1\rangle$态)

    Fig. 6.  Population dynamics of five-level double-Λ system fot $\varDelta_1=590\pi$ MHz, $\varDelta_2=\Delta_3=600\pi$ MHz, $\varDelta_4=610\pi$ MHz and $\varOmega_1=\varOmega_2=\varOmega_3=\varOmega_4=120\pi$ MHz starting from the initial state $|\psi_0\rangle=|-1\rangle$. The solid line represents the population before elimination (blue, green, orange, purple, and gray line represent levels $|1\rangle $, $|0\rangle $, $|-1\rangle $, $|a\rangle $, $ |b\rangle$, respectively), and the circular dot represents the population of the three energy levels after elimination (blue, green, and orange dots correspond to levels $|1\rangle $, $|0\rangle $, $|-1\rangle $, respectively)

    图 7  双V型五能级系统能级图

    Fig. 7.  Five-level double-V type hermitian system energy level diagram

    图 8  双V型五能级系统绝热消除后的数值解布居数动力学图. 单光子失谐$ \varDelta_1 = 590\pi \;{\rm{MHz}}$, $ \varDelta_2 = 595\pi \;{\rm{MHz}}$, $ \varDelta_3 = $$ 605\pi \;{\rm{MHz}}$, $ \varDelta_4 = 610\pi \;{\rm{MHz}}$, 拉比频率$ \varOmega_1 = \varOmega_2 = \varOmega_3 = $$ \varOmega_4 = 120\pi \;{\rm{MHz}}$, 初态$ |\psi_0\rangle = |g_1\rangle $. 其中实线为消除前能级布居数(绿线、蓝线、橙线、灰线、浅蓝色线分别代表$ |g_1\rangle $, $ |g_2\rangle $, $ |e_1\rangle $, $ |e_2\rangle $, $ |e_3\rangle $态), 圆点为消除后有效二能级的布居数(绿点和蓝点分别代表$ |g_1\rangle $$ |g_2\rangle $态)

    Fig. 8.  Population dynamics of the five-level double-V system for $ \varDelta_1 = 590\pi \;{\rm{MHz}}$, $ \varDelta_2 = 595\pi \;{\rm{MHz}}$, $ \varDelta_3 = $$ 605\pi \;{\rm{MHz}}$, $ \varDelta_4 = 610\pi \;{\rm{MHz}}$ and $ \varOmega_1 = \varOmega_2 = $$ \varOmega_3 = \varOmega_4 = 120\pi \;{\rm{MHz}}$ starting from the initial state $ |\psi_0\rangle = |g_1\rangle $. Green, blue, orange, gray, light blue line denote the population of $|g_1\rangle $, $ |g_2\rangle$, $|e_1\rangle $, $|e_2\rangle $, $|e_3\rangle $ state before adiabatic elimination, respectively. Green and blue dots denote the population of $|g_1\rangle $ and $ |g_2\rangle$ state after adiabatic elimination, respectively.

  • [1]

    Kittel C 1963 Quantum Theory of Solids (New York: John Wiley and Sons

    [2]

    Goldman N, Juzeliūnas G, Öhberg P, Spielman I B 2014 Rep. Prog. Phys. 77 126401Google Scholar

    [3]

    Zheng W, Yu Z Q, Cui X L, Zhai H 2013 J. Phys. B 46 134007Google Scholar

    [4]

    Jian C M, Zhai H 2011 Phys. Rev. B. 84 060508Google Scholar

    [5]

    Zhang J Y, Ji S C, Chen Z, Zhang L, Du Z D, Yan B, Pan G S, Zhao B, Deng Y J, Zhai H, Chen S, Pan J W 2012 Phys. Rev. Lett. 109 115301Google Scholar

    [6]

    焦宸, 简粤, 张爱霞, 薛具奎 2023 物理学报 72 060302Google Scholar

    Jiao C, Jian Y, Zhang A X, Xue J K 2023 Acta Phys. Sin. 72 060302Google Scholar

    [7]

    施婷婷, 汪六九, 王璟琨, 张威 2020 物理学报 69 016701Google Scholar

    Shi T T, Wang L J, Wang J K, Zhang W 2020 Acta Phys. Sin. 69 016701Google Scholar

    [8]

    陈星, 薛潇博, 张升康, 马余全, 费鹏, 姜元, 葛军 2021 物理学报 70 083401Google Scholar

    Chen X, Xue X B, Zhang S K, Ma Y Q, Fei P, Jiang Y, Ge J 2021 Acta Phys. Sin. 70 083401Google Scholar

    [9]

    Campbell D L, Price R M, Putra A, Valdés-Curiel A, Trypogeorgos D, Spielman I B 2016 Nat. Commun. 7 10897Google Scholar

    [10]

    Galitski V, Spielman I B 2013 Nature. 494 49Google Scholar

    [11]

    Stanescu T D, Anderson B, Galitski V 2008 Phys. Rev. A. 78 023616Google Scholar

    [12]

    Xu P, Deng T S, Zheng W, Zhai H 2015 Phys. Rev. A 103 L061302Google Scholar

    [13]

    Wang P J, Yu Z Q, Fu Z k, Miao J, Huang L H, Chai S J, Zhai H, Zhang J 2012 Phys. Rev. Lett. 109 095301Google Scholar

    [14]

    Cheuk L W, Sommer A T, Hadzibabic Z, Yefsah T, Bakr W S, Zwierlein M W 2012 Phys. Rev. Lett. 109 095302Google Scholar

    [15]

    Chen X L, Wang J, Li Y, Liu X J, Hu H 2018 Phys. Rev. A. 98 013614Google Scholar

    [16]

    Lan Z H, Öhberg P 2014 Phys. Rev. A. 89 023630Google Scholar

    [17]

    Chen L, Pu H, Zhang Y B 2016 Phys. Rev. A. 93 013629Google Scholar

    [18]

    Zhai H 2012 Int. J. Mod. Phys. B 26 1230001Google Scholar

    [19]

    Zhai H 2015 Rep. Prog. Phys. 78 026001Google Scholar

    [20]

    贺丽, 余增强 2017 物理学报 66 220301Google Scholar

    He L, Yu Z Q 2017 Acta Phys. Sin. 66 220301Google Scholar

    [21]

    Geier K T, Martone G I, Hauke P, Stringari S 2021 Phys. Rev. Lett. 127 115301Google Scholar

    [22]

    Liao R Y 2018 Phys. Rev. Lett. 120 140403Google Scholar

    [23]

    Chen L, Zhang Y B, Pu H 2020 Phys. Rev. Lett. 125 195303Google Scholar

    [24]

    Li Y, Pitaevskii L P, Stringari S 2012 Phys. Rev. Lett. 108 225301Google Scholar

    [25]

    Lin Y J, Jiménez-García K, Spielman I B 2011 Nature. 471 83Google Scholar

    [26]

    Steck D A 2007 Quantum and Atom Optics (Eugene, Oregon: Department of Physics, University of Oregon

    [27]

    Dalibard J, Gerbier F, Juzeliūnas G, Öhberg P 2011 Rev. Mod. Phys. 83 1523Google Scholar

    [28]

    Bergmann K, Theuer H, Shore B W 1998 Rev. Mod. Phys. 70 1003Google Scholar

    [29]

    Král P, Thanopulos I, Shapiro M 2007 Rev. Mod. Phys. 79 53Google Scholar

    [30]

    Brion E, Pedersen L H, Mølmer K 2007 J. Phys. A 40 1033Google Scholar

    [31]

    Li H, Shen H Z, Wu S L, Yi X X 2017 Optics Express. 25 30135Google Scholar

    [32]

    Li G Q, Chen G D, Peng P, Qi W 2017 Eur. Phys. J. D 71 14Google Scholar

    [33]

    Guo L P, Du L, Yin C H, Zhang Y B, Chen S 2018 Phys. Rev. A. 97 032109Google Scholar

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出版历程
  • 收稿日期:  2023-06-27
  • 修回日期:  2023-08-06
  • 上网日期:  2023-08-24
  • 刊出日期:  2023-11-05

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