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Anderson局域化是凝聚态物理中一个影响深远的现象, 它代表了由无序引发的本征态的根本性转变. 本文提出了一个基于超冷原子动量态晶格系统的实验方案, 用以实现弯曲时空下的Aubry-André-Harper (AAH) 模型, 并研究其中的Anderson局域化. 得益于每对相邻动量态之间耦合的单独可操控性, 动量态晶格中的耦合强度可以被编辑成幂律位置依赖的形式$J_n \propto n^{\sigma}$, 从而能够有效模拟弯曲时空. 动量态晶格中波包演化的数值计算结果表现出初始格点依赖的局域化性质, 符合理论预测的相分离现象. 通过分析波包演化动力学数据, 可以观测到相分离临界格点的移动. 同时, 本文还提出了通过调制时空弯曲参数σ来制备本征态的方案, 并在动量态晶格中进行了数值仿真. 最后, 在不同准周期调制相位下制备能谱中所有本征态, 分析了本征态的局域化性质, 验证了在能谱中共存的局域相、延展相和摇摆相. 本文为在实验中研究弯曲时空下的Anderson局域化物理提供了新的可行途径.
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关键词:
- Anderson局域化 /
- 弯曲时空 /
- 超冷原子 /
- 动量态晶格
Anderson localization is a profound phenomenon in condensed matter physics, representing a fundamental transition in eigenstates, which is triggered off by disorder. The one-dimensional Aubry-André-Harper (AAH) model, an iconic quasiperiodic lattice model, is one of the simplest models that demonstrate the Anderson localization transition. Recently, with the growth of interest in quantum lattice models in curved spacetime (CST), the AAH model in CST has been proposed to explore the interplay between Anderson localization and CST physics. Several CST lattice models have been realized in optical waveguide systems to date, but there are still significant challenges to the experimental preparation and measurement of states, primarily due to the difficulty in dynamically modulating the lattices in such systems. In this work, we propose an experimental scheme using a momentum-state lattice (MSL) in an ultracold atom system to realize the AAH model in CST and study the Anderson localization in this context. Due to the individually controllable coupling between adjacent momentum states in each pair, the coupling amplitude in the MSL can be encoded as a power-law position-dependent $J_n \propto n^{\sigma}$, which is conducive to the effective simulation of CST. The numerical calculation results of the MSL Hamiltonian show that the phase separation appears in a 34-site AAH chain in CST, where wave packet dynamics exhibit the localized behavior on one side of the critical site and the extended behavior on the other side. The critical site of phase separation is identified by extracting the turning points of the evolving fractal dimension and wave packet width from the evolution simulations. Furthermore, by modulating the spacetime curvature parameter σ, we propose a method of preparing the eigenstates of the AAH chain in CST, and perform numerical simulations in the MSL. By calculating the fractal dimension of eigenstates prepared using the aforementioned method, we analyze the localization properties of eigenstates under various quasiperiodic modulation phases, confirming the coexistence of localized phase, swing phase, and extended phase in the energy spectrum. Unlike traditional localized and extended phases, eigenstates in the swing phase of the AAH model in CST exhibit different localization properties under different modulation phases, indicating the existence of a swing mobility edge. Our results provide a feasible experimental method for studying Anderson localization in CST and presents a new platform for realizing quantum lattice models in curved spacetime.-
Keywords:
- Anderson localization /
- curved spacetime /
- ultracold atoms /
- momentum-state lattice
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Google Scholar
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Google Scholar
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图 1 (a)平坦时空下的AAH模型示意图, 其中$ \sigma=0 $, 跃迁均为J; (b)弯曲时空下的AAH模型示意图, 其中$ \sigma \geqslant 1 $, 跃迁$ J_n $是幂律格点位置依赖的; (c) $ \sigma=0 $的平坦时空和(d) $ \sigma=1 $的弯曲时空下AAH链本征态的分形维数$ \varGamma $作为本征能量$ E/J $和调制强度$ \lambda/J $的函数. 图(c)中$ \phi=0 $, 图(d)中数据选取了$0— 2{\text{π}}$一共$ 41 $个ϕ的值进行计算并取平均. 其他参数设置为L = F15=610
Fig. 1. (a) Schematic of AAH model in flat spacetime, where $ \sigma=0 $ and all hoppings are J. (b) Schematic of AAH model in curved spacetime, where $ \sigma \geqslant 1 $ and hopping $ J_n $ is power-law site position dependent. Fractal dimensions $ \varGamma $ of eigenstates of AAH chain in (c) flat spacetime with $ \sigma=0 $ and (d) curved spacetime with $ \sigma=1 $ as a function of the eigenenergy $ E/J $ and modulation amplitude $ \lambda/J $. In panel (c) $ \phi=0 $, and data in panel (d) are calculated and averaged over $ 41 $ values of ϕ ranging from $ 0 $ to $ 2{\text{π}} $. Other parameters: $ L=F_{15}=610 $.
图 2 (a)动量态晶格示意图; (b)弯曲时空下AAH链中的波包演化动力学. 从左至右$ \lambda/J $分别取$ 0 $, $ 1.0 $, $ 1.5 $和$ 3.0 $, 上图和下图的波包初始格点分别为$ n_0 = 10 $和$ n_0=30 $. 灰色虚线标记了相分离临界格点$ n_{\rm{c}} $的理论值, 其他参数设置: $ \sigma = 1 $, $ \phi=0 $
Fig. 2. (a) Schematic of momentum-state lattice; (b) wave packet evolution dynamics in AAH chain in curved spacetime. From left to right, $ \lambda/J = 0 $, $ 1.0 $, $ 1.5 $, and $ 3.0 $. The initial sites of wave packet are $ n_0 = 10 $ (top panel) and $ n_0 = 30 $ (bottom panel), respectively. The grey dashed line marks the theoretical value of the phase separation critical site $ n_{\rm{c}} $. Other parameters: $ \sigma = 1 $ and $ \phi=0 $.
图 3 观测相分离临界格点. 不同格点能量调制强度$ \lambda/J $下, (a)时间平均的演化分形维数$ \overline{\varGamma}_{\rm{evo}} $和(b)时间平均的波包宽度$ \overline{W} $作为波包初始格点位置的函数, 其中时间平均的范围为$ 1\;{\mathrm{ms}} $总演化时间的最后$ 0.5\;{\mathrm{ms }}$. (c)相分离临界格点$ n_{\rm{c}} $作为$ \lambda/J $的函数, 其中红色菱形和蓝色菱形分别从$ \overline{\varGamma}_{\rm{evo}} $和$ \overline{W} $数据中计算得到, 黑线是理论计算值. 其他参数设置: $ \sigma = 1 $以及$ \phi=0 $
Fig. 3. Observation of the phase separation critical site: (a) The time-averaged evolving fractal dimension $ \overline{\varGamma}_{\rm{evo}} $ and (b) the time-averaged wave packet width $ \overline{W} $ as a function of the initial site for various $ \lambda/J $, where the time averaging range covers the final $ 0.5\;{\mathrm{ms }} $ of the total evolution time of $ 1\;{\mathrm{ms }} $; (c) the phase separation critical site $ n_{\rm{c}} $ as a function of $ \lambda/J $. The red and blue diamonds are obtained from the data of $ \overline{\varGamma}_{\rm{evo}} $ and $ \overline{W} $, respectively, and the black line represents the theoretical value. Other parameters: $ \sigma = 1 $ and $ \phi=0 $.
图 4 本征能量(a) $ E/J = -2.131 $和(b) $ E/J = 0.122 $的本征态制备过程的能谱, 其中蓝色点为制备路径, 红色点为同一时刻下与制备路径上本征态之间绝热参数$ A > 0.1 $的本征态; (c)和(d)分别为(a)和(b)中对应的制备保真度, 插图中蓝色柱状图分别是零时刻的波包分布和保持阶段的时间平均波包分布, 虚线柱状图为对应的理论本征态. 制备参数: (a) $ n_0 = 33 $, $ \alpha = 3 $, $ t_{\rm{evo}} = $$ 0.80 $ms以及$ \sigma= 30 \to 1 $; (b) $ n_0 = 23 $, $ \alpha = 5 $, $ t_{\rm{evo}} = 0.80 $ ms以及$ \sigma= 30 \to 1 $. 保持阶段(阴影区域)持续时间为$ t_{\rm{hold}} = $$ 0.25 $ ms, 其他参数设置: $ \lambda/J = 1.5 $以及$ \phi=0 $
Fig. 4. Energy spectrum of the preparation process of eigenstates with eigenenergy (a) $ E/J = -2.131 $ and (b) $ E/J = 0.122 $, where the blue points represent the preparation path, and red points mark eigenstates with an adiabatic parameter $ A > 0.1 $ with the eigenstate in the preparation path at the same time. (c) and (d) are the corresponding preparation fidelity in panels (a) and (b), respectively. In the insets, the blue bars indicate the wave packet distribution at zero time and the time-averaged wave packet distribution during the holding stage, while the dotted bars denote the corresponding theoretical eigenstates. Preparation parameters: (a) $ n_0 = 33 $, $ \alpha = 3 $, $ t_{\rm{evo}} = 0.80 $ ms, and $ \sigma= 30 \to 1 $; (b) $ n_0 = 23 $, $ \alpha = 5 $, $ t_{\rm{evo}} = 0.80 $ ms, and $ \sigma= 30 \to 1 $. The holding stage (shaded area) lasts for $ t_{\rm{hold}} = 0.25 $ ms. Other parameters: $ \lambda/J = 1.5 $ and $ \phi=0 $.
图 5 观测能谱中三种不同的相 (a)本征能量作为能级指标β的函数, 其中蓝色点为$ L_{\rm{loc}} = 24 $的局域子链, 红色点为$ L_{\rm{ext}} = $$ 10 $的延展子链, 时空弯曲参数$ \sigma=1 $; (b) $ \overline{\varGamma} $作为本征能量的函数, 其中阴影区域取自图(a). 数据点包含$ \phi = 0 $, $ 0.5\pi $, $ 1.0\pi $和$ 1.5\pi $四种相位, 制备过程中σ从$ 30 $动态调制到$ 1 $. 其他参数: $ \lambda/J = 1.5 $
Fig. 5. Observation of three distinct phase in the energy spectrum: (a) Eigenenergy as a function of energy level index. Here, the blue points denote the localized subchain with $ L_{\rm{loc}} = 24 $, and the red points represent the extended subchain with $ L_{\rm{ext}} = 10 $. The spacetime curvature parameter $ \sigma = 1 $; (b) $ \overline{\varGamma} $ as a function of eigenenergy, where the shaded area is taken from panel (a). The data consists of four phases $ \phi = 0 $, $ 0.5\pi $, $ 1.0\pi $ and $ 1.5\pi $, and σ is modulated from $ 30 $ to $ 1 $ during preparation. Other parameters: $ \lambda/J = 1.5 $.
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[1] Anderson P W 1958 Phys. Rev. 109 1492
Google Scholar
[2] Lee P A, Ramakrishnan T V 1985 Rev. Mod. Phys. 57 287
Google Scholar
[3] Evers F, Mirlin A D 2008 Rev. Mod. Phys. 80 1355
Google Scholar
[4] Mott N F 2001 Adv. Phys. 50 865
Google Scholar
[5] Billy J, Josse V, Zuo Z, Bernard A, Hambrecht B, Lugan P, Clément D, Sanchez-Palencia L, Bouyer P, Aspect A 2008 Nature 453 891
Google Scholar
[6] Kondov S S, McGehee W R, Zirbel J J, DeMarco B 2011 Science 334 66
Google Scholar
[7] Jendrzejewski F, Bernard A, Müller K, et al. 2012 Nat. Phys. 8 398
Google Scholar
[8] Semeghini G, Landini M, Castilho P, Roy S, Spagnolli G, Trenkwalder A, Fattori M, Inguscio M, Modugno G 2015 Nat. Phys. 11 554
Google Scholar
[9] Lahini Y, Avidan A, Pozzi F, Sorel M, Morandotti R, Christodoulides D N, Silberberg Y 2008 Phys. Rev. Lett. 100 013906
Google Scholar
[10] Sperling T, Bührer W, Aegerter C M, Maret G 2013 Nat. Photonics 7 48
Google Scholar
[11] Wiersma D S 2013 Nat. Photonics 7 188
Google Scholar
[12] Yu S, Qiu C W, Chong Y, Torquato S, Park N 2021 Nat. Rev. Mater. 6 226
Google Scholar
[13] Abrahams E, Anderson P W, Licciardello D C, Ramakrishnan T V 1979 Phys. Rev. Lett. 42 673
Google Scholar
[14] Das Sarma S, Kobayashi A, Prange R E 1986 Phys. Rev. Lett. 56 1280
Google Scholar
[15] Das Sarma S, He S, Xie X C 1990 Phys. Rev. B 41 5544
Google Scholar
[16] Biddle J, Das Sarma S 2010 Phys. Rev. Lett. 104 070601
Google Scholar
[17] Ganeshan S, Pixley J H, Das Sarma S 2015 Phys. Rev. Lett. 114 146601
Google Scholar
[18] Yao H, Khoudli A, Bresque L, Sanchez-Palencia L 2019 Phys. Rev. Lett. 123 070405
Google Scholar
[19] Aubry S, André G 1980 Ann. Isr. Phys. Soc. 3 18
[20] Deng X, Ray S, Sinha S, Shlyapnikov G V, Santos L 2019 Phys. Rev. Lett. 123 025301
Google Scholar
[21] Danieli C, Bodyfelt J D, Flach S 2015 Phys. Rev. B 91 235134
Google Scholar
[22] Li X, Li X, Das Sarma S 2017 Phys. Rev. B 96 085119
Google Scholar
[23] Wang Y, Xia X, Zhang L, Yao H, Chen S, You J, Zhou Q, Liu X J 2020 Phys. Rev. Lett. 125 196604
Google Scholar
[24] Lüschen H P, Scherg S, Kohlert T, Schreiber M, Bordia P, Li X, Das Sarma S, Bloch I 2018 Phys. Rev. Lett. 120 160404
Google Scholar
[25] An F A, Padavić K, Meier E J, Hegde S, Ganeshan S, Pixley J H, Vishveshwara S, Gadway B 2021 Phys. Rev. Lett. 126 040603
Google Scholar
[26] Wang Y, Zhang J H, Li Y, Wu J, Liu W, Mei F, Hu Y, Xiao L, Ma J, Chin C, Jia S 2022 Phys. Rev. Lett. 129 103401
Google Scholar
[27] HAWKING S W 1974 Nature 248 30
Google Scholar
[28] Unruh W G 1976 Phys. Rev. D 14 870
Google Scholar
[29] Unruh W G 1981 Phys. Rev. Lett. 46 1351
Google Scholar
[30] Hu J, Feng L, Zhang Z, Chin C 2019 Nat. Phys. 15 785
Google Scholar
[31] Muñoz de Nova J R, Golubkov K, Kolobov V I, Steinhauer J 2019 Nature 569 688
Google Scholar
[32] Drori J, Rosenberg Y, Bermudez D, Silberberg Y, Leonhardt U 2019 Phys. Rev. Lett. 122 010404
Google Scholar
[33] Almeida C R, Jacquet M J 2023 Eur. Phys. J. H 48 15
Google Scholar
[34] Kedem Y, Bergholtz E J, Wilczek F 2020 Phys. Rev. Res. 2 043285
Google Scholar
[35] Morice C, Moghaddam A G, Chernyavsky D, van Wezel J, van den Brink J 2021 Phys. Rev. Res. 3 L022022
Google Scholar
[36] Sheng C, Huang C, Yang R, Gong Y, Zhu S, Liu H 2021 Phys. Rev. A 103 033703
Google Scholar
[37] Mertens L, Moghaddam A G, Chernyavsky D, Morice C, van den Brink J, van Wezel J 2022 Phys. Rev. Res. 4 043084
Google Scholar
[38] Könye V, Morice C, Chernyavsky D, Moghaddam A G, van den Brink J, van Wezel J 2022 Phys. Rev. Res. 4 033237
Google Scholar
[39] Li S Z, Yu X J, Zhu S L, Li Z 2023 Phys. Rev. B 108 094209
Google Scholar
[40] Wang Y, Sheng C, Lu Y H, et al. 2020 Natl. Sci. Rev. 7 1476
Google Scholar
[41] He R, Zhao Y, Sheng C, Duan J, Wei Y, Sun C, Lu L, Gong Y X, Zhu S, Liu H 2024 Phys. Rev. Res. 6 013233
Google Scholar
[42] Fort C, Zaccanti M, Modugno G, Modugno M, Inguscio M 2008 Nature 453 895
Google Scholar
[43] Schreiber M, Hodgman S S, Bordia P, Lüschen H P, Fischer M H, Vosk R, Altman E, Schneider U, Bloch I 2015 Science 349 842
Google Scholar
[44] Parshin D A, Schober H R 1999 Phys. Rev. Lett. 83 4590
Google Scholar
[45] Zhou X C, Wang Y, Poon T F J, Zhou Q, Liu X J 2023 Phys. Rev. Lett. 131 176401
Google Scholar
[46] Meier E J, An F A, Dauphin A, Maffei M, Massignan P, Hughes T L, Gadway B 2018 Science 362 929
Google Scholar
[47] Li H, Dong Z, Longhi S, Liang Q, Xie D, Yan B 2022 Phys. Rev. Lett. 129 220403
Google Scholar
[48] Yuan T, Zeng C, Mao Y Y, Wu F F, Xie Y J, Zhang W Z, Dai H N, Chen Y A, Pan J W 2023 Phys. Rev. Res. 5 L032005
Google Scholar
[49] Zeng C, Shi Y R, Mao Y Y, et al. 2024 Phys. Rev. Lett. 132 063401
Google Scholar
[50] Gadway B 2015 Phys. Rev. A 92 043606
Google Scholar
[51] Xiao T, Xie D, Gou W, Chen T, Deng T S, Yi W, Yan B 2020 Eur. Phys. J. D 74 152
Google Scholar
[52] Khaykovich L, Schreck F, Ferrari G, Bourdel T, Cubizolles J, Carr L D, Castin Y, Salomon C 2002 Science 296 1290
Google Scholar
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