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在拓扑系统中, 探索相互作用引起的新奇的拓扑泵浦现象日益受到人们的关注, 其中包括由相互作用诱导的非线性拓扑泵浦. 本文提出可以利用超冷原子动量光晶格系统, 有效地模拟一维非线性的非对角Aubry-André-Harper (AAH) 模型, 研究非线性拓扑泵浦的实验方案. 首先, 通过数值方法计算了一维非对角AAH模型的非线性能带结构随相互作用强度的变化, 得到了非线性系统的孤子态解. 然后, 分析了不同相互作用强度下孤子态的拓扑输运, 发现其质心的移动距离具有量子化的输运特征, 由所占据能带的陈数决定, 并讨论了非线性拓扑泵浦对相互作用符号的依赖性. 同时还计算了在不同相互作用强度下, 系统最低能带和最高能带对应陈数的分布. 最后, 基于
$ ^{7}\text{Li}$ 原子的动量光晶格实验系统, 提出了一个非线性拓扑泵浦方案. 本文构造了一种近似于孤子态分布的初始态并演示了其动力学演化过程, 并分析了绝热演化条件对泵浦过程的影响. 结果表明, 在动量晶格系统中演示非线性拓扑泵浦具有可行性. 本文的工作为在超冷原子系统中研究非线性拓扑泵浦提供了一个可行的途径, 有助于进一步探测非线性引起的拓扑相变和边界效应.Topological pumping enables the quantized transport of matter waves through an adiabatic evolution of the system, which plays an essential role in the applications of transferring quantum states and exploring the topological properties in higher-dimensional quantum systems. Recently, exploring the interplay between novel topological pumping and interactions has attracted growing attention in topological systems, such as nonlinear topological pumping induced by interactions. So far, the experimental realizations of the nonlinear topological pumps have been realized only in the optical waveguide systems with Kerr nonlinearity. It is still necessary to further explore the phenomenon in different systems. Here, we present an experimental proposal for realizing the nonlinear topological pumping via a one-dimensional (1D) off-diagonal Aubry-André-Harper (AAH) model with mean-field interactions in the momentum space lattice of ultracold atoms. In particular, we develop a numerical method for calculating the energy band of the nonlinear systems. With numerical calculations, we first solve the nonlinear energy band structure and soliton solution of the 1D nonlinear off-diagonal AAH model in the region of weak interaction strengths. The result shows that the lowest or the highest energy band is modulated in the nonlinear system of$ g > 0$ or$ g < 0$ , respectively. The eigenstates of the associated energy bands have the features of the soliton solutions. We then show that the topological pumping of solitons exhibits quantized transport characteristics. Moreover, we numerically calculate the Chern number associated with the lowest and highest energy bands at different interaction strengths. The result shows that the quantized transport of solitons is determined by the Chern number of the associated energy band of the system from which solitons emanate. Finally, we propose a nonlinear topological pumping scheme based on a momentum lattice experimental system with$ ^{7}\text{Li}$ atoms. We can prepare the initial state, which is approximately the distribution of the soliton state of the lowest energy band, and calculate the dynamical evolution of this initial state in the case of$ U > 0$ . Also, we analyze the influence of adiabatic evolution conditions on the pumping process, demonstrating the feasibility of nonlinear topological pumping in the momentum lattice system. Our study provides a feasible route for investigating nonlinear topological pumping in ultracold atom systems, which is helpful for further exploring the topological transport in nonlinear systems, such as topological phase transitions and edge effects induced by nonlinearity.[1] Thouless D J 1983 Phys. Rev. B 27 6083
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[2] Nakajima S, Tomita T, Taie S, Ichinose T, Ozawa H, Wang L, Troyer M, Takahashi Y 2016 Nat. Phys. 12 296
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[3] Hu S, Ke Y G, Lee C H 2020 Phys. Rev. A 101 052323
Google Scholar
[4] Lohse M, Schweizer C, Price H M, Zilberberg O, Bloch I 2018 Nature 553 55
Google Scholar
[5] Citro R, Aidelsburger M 2023 Nat. Rev. Phys. 5 87
Google Scholar
[6] Cerjan A, Wang M, Huang S, Chen K P, Rechtsman M C 2020 Light Sci. Appl. 9 178
Google Scholar
[7] Ke Y G, Qin X Z, Kivshar Y S, Lee C H 2017 Phys. Rev. A 95 063630
Google Scholar
[8] Lohse M, Schweizer C, Zilberberg O, Aidelsburger M, Bloch I 2016 Nat. Phys. 12 350
Google Scholar
[9] Ma W, Zhou L, Zhang Q, Li M, Cheng C, Geng J, Rong X, Shi F, Gong J, Du J 2018 Phys. Rev. Lett. 120 120501
Google Scholar
[10] Zilberberg O, Huang S, Guglielmon J, Wang M, Chen K P, Kraus Y E, Rechtsman M C 2018 Nature 553 59
Google Scholar
[11] Cheng W, Prodan E, Prodan C 2020 Phys. Rev. Lett. 125 224301
Google Scholar
[12] Jung P S, Parto M, Pyrialakos G G, et al. 2022 Phys. Rev. A 105 013513
Google Scholar
[13] Fu Q, Wang P, Kartashov Y V, Konotop V V, Ye F 2022 Phys. Rev. Lett. 128 154101
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[14] Mostaan N, Grusdt F, Goldman N 2022 Nat. Commun. 13 5997
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[15] Jürgensen M, Mukherjee S, Rechtsman M C 2021 Nature 596 63
Google Scholar
[16] Jürgensen M, Mukherjee S, Jörg C, Rechtsman M C 2023 Nat. Phys. 19 420
Google Scholar
[17] Schäfer F, Fukuhara T, Sugawa S, Takasu Y, Takahashi Y 2020 Nat. Rev. Phys. 2 411
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[18] Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225
Google Scholar
[19] Kartashov Y V, Malomed B A, Torner L 2011 Rev. Mod. Phys. 83 247
Google Scholar
[20] Kevrekidis P G, Frantzeskakis D J, Carretero-González R 2008 Emergent Nonlinear Phenomena in Bose-Einstein Condensates (Vol. 45) (Berlin: Springer) pp99–130
[21] Gadway B 2015 Phys. Rev. A 92 043606
Google Scholar
[22] An F A, Sundar B, Hou J, Luo X W, Meier E J, Zhang C, Hazzard K R A, Gadway B 2021 Phys. Rev. Lett. 127 130401
Google Scholar
[23] An F A, Padavicć K, Meier E J, Hegde S, Ganeshan S, Pixley J H, Vishveshwara S, Gadway B 2021 Phys. Rev. Lett. 126 040603
Google Scholar
[24] Aubry S, André G 1980 Ann. Isr. Phys. Soc. 3 133
[25] Harper P G 1955 Proc. Phys. Soc. A 68 874
Google Scholar
[26] Cao J, Xing Y, Qi L, Wang D Y, Bai C H, Zhu A D, Zhang S, Wang H F 2018 Laser Phys. Lett. 15 015211
Google Scholar
[27] Martinez Alvarez V M, Coutinho-Filho M D 2019 Phys. Rev. A 99 013833
Google Scholar
[28] Nguyen J H V, Dyke P, Luo D, Malomed B A, Hulet R G 2014 Nat. Phys. 10 918
Google Scholar
[29] Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959
Google Scholar
[30] Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Jpn. 74 1674
Google Scholar
[31] Strecker K E, Partridge G B, Truscott A G, Hulet R G 2002 Nature 417 150
Google Scholar
[32] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr L D, Castin Y, Salomon C 2002 Science 296 1290
Google Scholar
[33] Leykam D, Chong Y D 2016 Phys. Rev. Lett. 117 143901
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[34] Bongiovanni D, Jukić D, Hu Z, Lunić F, Hu Y, Song D, Morandotti R, Chen Z, Buljan H 2021 Phys. Rev. Lett. 127 184101
Google Scholar
[35] Kartashov Y V, Arkhipova A A, Zhuravitskii S A, Skryabin N N, Dyakonov I V, Kalinkin A A, Kulik S P, Kompanets V O, Chekalin S V, Torner L, Zadkov V N 2022 Phys. Rev. Lett. 128 093901
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图 1 (a)一维非对角AAH模型示意图, 每个晶格原胞有3个格点(A, B, C), 最近邻格点之间的耦合强度($J_{{ab}}$, $J_{{bc}}$, $J_{{ca}}$)随时间变化, 每个格点上的能量设置为零; (b)在一个泵浦周期内, 耦合强度的周期性调制函数 (由(2)式定义); (c)在$\varOmega t=0$时刻, 非对角AAH模型非线性能带结构在不同相互作用强度$g = 0, 1.5, 2.0$下的分布. 图中的物理量均以$J_{{\rm{max}}}$为单位, 耦合强度值$J_{{ab}}=0.77$, $J_{{bc}}=0.10$和$J_{{ca}}=0.77$
Fig. 1. (a) Schematic illustration of 1D off-diagonal AAH model with three sites (A, B, C) per unit cell and time-dependent couplings ($J_{{ab}}$, $J_{{bc}}$, $J_{{ca}}$) between neighbouring sites; (b) variation of the couplings during one pumping cycle defined by Eq. (2); (c) energy bands of nonlinear off-diagonal AAH model vs. interaction strength g. All quantities shown in the pictures are given in units of $J_{{\rm{max}}}$, with coupling strength values $J_{{ab}}=0.77$, $J_{{bc}}=0.10$ and $J_{{ca}}=0.77$
图 2 (a), (b)在$g > 0$的系统中, 计算得到的最低能带的孤子态的波函数分布, 在淬火动力学演化过程中是严格局域化的; (c), (d)在$g<0$的系统中, 最高能带的孤子态波函数分布和能带结构的分布
Fig. 2. (a), (b) In the system of $g > 0$, the wave function distribution of soliton state for the lowest energy band is strictly localized in the process of the quench dynamics; (c), (d) in the system of $g < 0$, the wave function distribution of soliton state for the highest band and the energy band structure
图 3 非线性拓扑泵浦 (a) $g=0$系统中, 最低能带上分布的瓦尼尔态的线性泵浦演化; (b) $g=1.5$系统中, 最低能带的孤子态的非线性演化; (c) $g=-1.5$系统中, 最高能带的孤子态的非线性演化; (d) 在两个泵浦周期内, 系统的质心位移结果. 上述结果均是对(3)式进行数值求解所得, 所用参数: (a)耦合强度$J_{{\rm{max}}} =1$, 调制频率$\varOmega / J_{{\rm{max}}}=0.02$, 原胞数$N_{\rm{c}} =101$; (b), (c)耦合强度$J_{{\rm{max}}} =1$, 调制频率$\varOmega / J_{{\rm{max}}} =0.01$, 原胞数$N_{\rm{c}}=21$
Fig. 3. Interaction induced nonlinear propagation in topological pumps: (a) At $g =0$, the linear pump evolution of uniformly distributed Wanier states at the lowest band; (b) at $g =1.5$, the nonlinear evolution of the soliton state for the lowest occupancy band; (c) at the $g = -1.5$, the nonlinear evolution of the soliton state for the highest occupancy band; (d) displacement of the centre of mass for the cases shown in a to c. The results are obtained by numerically solving Eq. (3) with parameters: (a) $J_{{\rm{max}}} =1$, $\varOmega / J_{{\rm{max}}} = 0.02$, $N_{\rm{c}} = 101$; (b), (c)$J_{{\rm{max}}} = 1$, $\varOmega / J_{{\rm{max}}} = 0.01$, $N_{\rm{c }}= 21$
图 4 在$g > 0$和$g < 0$的弱相互作用系统中, 分别计算最低能带的陈数$C_0$和最高能带的陈数$C_2$
Fig. 4. Chern number associated with the energy band are calculated for $g > 0$ and $g < 0$ in the regime of weak interaction strengths, respectively
图 5 利用动量晶格系统演示非线性拓扑泵浦方案 (a) 动量晶格示意图; (b) 初态制备过程; (c) 在两个泵浦周期内, 调制频率为$\varOmega/J_{{\rm{max}}} = 0.5$时孤子态的动力学演化; (d) 在两个泵浦周期内质心移动的晶格距离(红线为动量晶格的实际哈密顿量计算的结果, 蓝线为理想哈密顿量计算的结果); (e) 绝热演化条件分析, 不同$\varOmega/J_{{\rm{max}}}$对应的每个泵浦周期质心位置的移动距离. 设置参数为: $J_{{\rm{max}}} = 2 \pi \times 10.0$ kHz, $U / J_{{\rm{max}}} = 1.5$, $N_{\rm{c}} = 21$
Fig. 5. Implementatial proposal of nonlinear topological pumping based on the momentum lattice: (a) Schematic diagram of the momentum lattice; (b) preparation of initial state; (c) dynamics evolution of soliton state in two pumping periods, with modulation frequency of $\varOmega/J_{{\rm{max}}} = 0.5$; (d) lattice displacement of the center-of-mass during two pumping periods (The red line is the result calculated from actual Hamiltonian of the momentum lattice, and the blue line is the result of the ideal Hamiltonian); (e) analysis of adiabatic evolution conditions. The shift of center-of-mass for each pumping period corresponding to different values of $\varOmega/J_{{\rm{max}}}$. Parameters are: $J_{{\rm{max}}} = 2 \pi \times 10.0$ kHz, $U / J_{{\rm{max}}} = 1.5$, $N_{\rm{c}} = 21$.
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[1] Thouless D J 1983 Phys. Rev. B 27 6083
Google Scholar
[2] Nakajima S, Tomita T, Taie S, Ichinose T, Ozawa H, Wang L, Troyer M, Takahashi Y 2016 Nat. Phys. 12 296
Google Scholar
[3] Hu S, Ke Y G, Lee C H 2020 Phys. Rev. A 101 052323
Google Scholar
[4] Lohse M, Schweizer C, Price H M, Zilberberg O, Bloch I 2018 Nature 553 55
Google Scholar
[5] Citro R, Aidelsburger M 2023 Nat. Rev. Phys. 5 87
Google Scholar
[6] Cerjan A, Wang M, Huang S, Chen K P, Rechtsman M C 2020 Light Sci. Appl. 9 178
Google Scholar
[7] Ke Y G, Qin X Z, Kivshar Y S, Lee C H 2017 Phys. Rev. A 95 063630
Google Scholar
[8] Lohse M, Schweizer C, Zilberberg O, Aidelsburger M, Bloch I 2016 Nat. Phys. 12 350
Google Scholar
[9] Ma W, Zhou L, Zhang Q, Li M, Cheng C, Geng J, Rong X, Shi F, Gong J, Du J 2018 Phys. Rev. Lett. 120 120501
Google Scholar
[10] Zilberberg O, Huang S, Guglielmon J, Wang M, Chen K P, Kraus Y E, Rechtsman M C 2018 Nature 553 59
Google Scholar
[11] Cheng W, Prodan E, Prodan C 2020 Phys. Rev. Lett. 125 224301
Google Scholar
[12] Jung P S, Parto M, Pyrialakos G G, et al. 2022 Phys. Rev. A 105 013513
Google Scholar
[13] Fu Q, Wang P, Kartashov Y V, Konotop V V, Ye F 2022 Phys. Rev. Lett. 128 154101
Google Scholar
[14] Mostaan N, Grusdt F, Goldman N 2022 Nat. Commun. 13 5997
Google Scholar
[15] Jürgensen M, Mukherjee S, Rechtsman M C 2021 Nature 596 63
Google Scholar
[16] Jürgensen M, Mukherjee S, Jörg C, Rechtsman M C 2023 Nat. Phys. 19 420
Google Scholar
[17] Schäfer F, Fukuhara T, Sugawa S, Takasu Y, Takahashi Y 2020 Nat. Rev. Phys. 2 411
Google Scholar
[18] Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225
Google Scholar
[19] Kartashov Y V, Malomed B A, Torner L 2011 Rev. Mod. Phys. 83 247
Google Scholar
[20] Kevrekidis P G, Frantzeskakis D J, Carretero-González R 2008 Emergent Nonlinear Phenomena in Bose-Einstein Condensates (Vol. 45) (Berlin: Springer) pp99–130
[21] Gadway B 2015 Phys. Rev. A 92 043606
Google Scholar
[22] An F A, Sundar B, Hou J, Luo X W, Meier E J, Zhang C, Hazzard K R A, Gadway B 2021 Phys. Rev. Lett. 127 130401
Google Scholar
[23] An F A, Padavicć K, Meier E J, Hegde S, Ganeshan S, Pixley J H, Vishveshwara S, Gadway B 2021 Phys. Rev. Lett. 126 040603
Google Scholar
[24] Aubry S, André G 1980 Ann. Isr. Phys. Soc. 3 133
[25] Harper P G 1955 Proc. Phys. Soc. A 68 874
Google Scholar
[26] Cao J, Xing Y, Qi L, Wang D Y, Bai C H, Zhu A D, Zhang S, Wang H F 2018 Laser Phys. Lett. 15 015211
Google Scholar
[27] Martinez Alvarez V M, Coutinho-Filho M D 2019 Phys. Rev. A 99 013833
Google Scholar
[28] Nguyen J H V, Dyke P, Luo D, Malomed B A, Hulet R G 2014 Nat. Phys. 10 918
Google Scholar
[29] Xiao D, Chang M C, Niu Q 2010 Rev. Mod. Phys. 82 1959
Google Scholar
[30] Fukui T, Hatsugai Y, Suzuki H 2005 J. Phys. Soc. Jpn. 74 1674
Google Scholar
[31] Strecker K E, Partridge G B, Truscott A G, Hulet R G 2002 Nature 417 150
Google Scholar
[32] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr L D, Castin Y, Salomon C 2002 Science 296 1290
Google Scholar
[33] Leykam D, Chong Y D 2016 Phys. Rev. Lett. 117 143901
Google Scholar
[34] Bongiovanni D, Jukić D, Hu Z, Lunić F, Hu Y, Song D, Morandotti R, Chen Z, Buljan H 2021 Phys. Rev. Lett. 127 184101
Google Scholar
[35] Kartashov Y V, Arkhipova A A, Zhuravitskii S A, Skryabin N N, Dyakonov I V, Kalinkin A A, Kulik S P, Kompanets V O, Chekalin S V, Torner L, Zadkov V N 2022 Phys. Rev. Lett. 128 093901
Google Scholar
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