搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

一维准周期晶格的性质及应用

王玉成 刘雄军 陈澍

引用本文:
Citation:

一维准周期晶格的性质及应用

王玉成, 刘雄军, 陈澍

Properties and applications of one dimensional quasiperiodic lattices

Wang Yu-Cheng, Liu Xiong-Jun, Chen Shu
PDF
HTML
导出引用
  • 准周期晶格在冷原子领域被广泛研究, 它使得人们可以在一维或者二维系统里研究扩展到安德森局域的转变. 2008年, Inguscio研究组在冷原子系统里制备了一维准周期晶格, 并观测到了安德森局域化现象, 这极大地推动了准周期系统的理论和实验研究. 后来, Bloch研究组在制备的一维和二维准周期晶格中都观测到了多体局域的现象. 最近, 他们还在准周期晶格中成功观测到迁移率边以及存在迁移率边的系统的多体局域现象. 这些冷原子实验推动了多体局域以及迁移率边等方向的研究. 准周期晶格已经成为一个平台, 它对很多物理现象的影响正在被广泛研究, 并可以尝试在冷原子实验中观测到这种影响. 本文结合作者的一些相关工作, 对一维准周期晶格一些近期的研究进行了简要综述, 介绍了一些相关的重要的冷原子实验, 讨论了准周期晶格的一些重要性质, 以及它对一些物理现象(比如拓扑态)的影响.
    Quasiperiodic lattices have been widely studied in cold atoms, which make one study extended-Anderson localization transition in one-dimensional (1D) or two-dimensional (2D) systems. In 2008, Inguscio's group prepared one-dimensional quasiperiodic lattice in cold atomic system and observed Anderson localization, which promoted the theoretical and experimental studies of quasiperiodic systems. Later, Bloch's group observed the many body localization in their prepared 1D and 2D quasiperiodic lattices. Recently, they also successfully observed the mobility edge and many body localization in a system with mobility edge in quasiperiodic lattices. These cold atom experiments have promoted the researches of many body localization and mobility edge. Quasiperiodic lattices have become a platform and its effects on many physical phenomena are being extensively studied, which can be expectantly observed in cold atomic experiments. This paper is based on the authors' some related work and briefly review some recent studies on the 1D quasiperiodic lattices, including some important cold atom experiments, some important properties of the quasiperiodic lattices and their effects on some physical phenomena, such as topological states.
      通信作者: 王玉成, wangyc3@sustc.edu.cn ; 陈澍, schen@iphy.ac.cn
    • 基金项目: 国家重点研发计划(批准号: 2016YFA0301604, 2016YFA0300600, 2016YFA0302104)、国家自然科学基金(批准号: 11674301, 11574008, 11761161003, 11425419)、中国青年千人计划和中国科学院战略优先研究计划(批准号: XDB07020000)资助的课题.
      Corresponding author: Wang Yu-Cheng, wangyc3@sustc.edu.cn ; Chen Shu, schen@iphy.ac.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant Nos. 2016YFA0301604, 2016YFA0300600, 2016YFA0302104), the National Natural Science Foundation of China (Grant Nos. 11674301, 11574008, 11761161003, 11425419), the Thousand-Young-Talent Program of China, and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB07020000).
    [1]

    王义遒 2007 原子的激光冷却与陷俘(北京: 北京大学出版社)

    Wang Y Q 2007 Atomic Laser Cooling And Trapping (Beijing: Peking University Press) (in Chinese)

    [2]

    Stwalley W C 1976 Phys. Rev. Lett. 37 1628Google Scholar

    [3]

    Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885Google Scholar

    [4]

    Lin Y J, Jimenez-Garcia K, Spielman I B 2011 Nature 471 83Google Scholar

    [5]

    Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y J, Chen S, Liu X J, Pan J W 2016 Science 354 83Google Scholar

    [6]

    Zhang L, Liu X J 2018 arXiv:1806.05628

    [7]

    Cooper N R, Dalibard J, Spielman I B 2018 arXiv:1803.00249

    [8]

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

    [9]

    Hu J, Feng L, Zhang Z, Chin C, 2018 arXiv:1807.07504

    [10]

    Aubry S, André G 1980 Ann. Israel Phys. Soc 3 18

    [11]

    Das Sarma S, He S, Xie X C 1988 Phys. Rev. Lett. 61 2144Google Scholar

    [12]

    Das Sarma S, He S, Xie X C 1990 Phys. Rev. B 41 5544Google Scholar

    [13]

    Biddle J, Das Sarma S 2010 Phys. Rev. Lett. 104 070601Google Scholar

    [14]

    Biddle J, Priour Jr D J, Wang B, Das Sarma S 2011 Phys. Rev. B 83 075105Google Scholar

    [15]

    Roati G, D’Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, Inguscio M 2008 Nature 453 895Google Scholar

    [16]

    Cai X, Lang L J, Chen S, Wang Y 2013 Phys. Rev. Lett. 110 176403Google Scholar

    [17]

    DeGottardi W, Sen D, Vishveshwara S 2013 Phys. Rev. Lett. 110 146404Google Scholar

    [18]

    Wang J, Liu X J, Gao X, Hu H 2016 Phys. Rev. B 93 104504Google Scholar

    [19]

    Wang Y, Wang Y, Chen S 2016 Eur. Phys. J. B 89 254Google Scholar

    [20]

    Iyer S, Oganesyan V, Refael G, Huse D A 2013 Phys. Rev. B 87 134202Google Scholar

    [21]

    Wang Y, Hu H, Chen S 2016 Eur. Phys. J. B 89 77Google Scholar

    [22]

    Setiawan F, Deng D L, Pixley J H 2017 Phys. Rev. B 96 104205Google Scholar

    [23]

    Schreiber M, Hodgman S S, Bordia P, Lüschen H P, Fischer M H, Vosk R, Altman E, Schneider U, Bloch I 2015 Science 349 842Google Scholar

    [24]

    Anderson P W 1958 Phys. Rev. 109 1492Google Scholar

    [25]

    Thouless D J 1974 Phys. Rep. 13 93Google Scholar

    [26]

    Schreiber M 1985 J. Phys. C 18 2493Google Scholar

    [27]

    Hashimoto Y, Niizeki K, Okabe Y 1992 J. Phys. A 25 5211Google Scholar

    [28]

    Modugno M 2009 New J. Phys. 11 033023Google Scholar

    [29]

    郎利君 2014 博士学位论文(北京: 中国科学院大学)

    Lang L J 2014 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese)

    [30]

    Li X, Li X, Das Sarma S 2017 Phys. Rev. B 96 085119Google Scholar

    [31]

    Lüschen H P, Scherg S, Kohlert T, Schreiber M, Bordia P, Li X, Das Sarma S, Bloch I 2018 Phys. Rev. Lett. 120 160404Google Scholar

    [32]

    Basko D, Aleiner I, Altshuler B 2006 Ann. Phys. 321 1126Google Scholar

    [33]

    Nandkishore R, Huse D A 2015 Annu. Rev. Condens. Matter Phys. 6 15Google Scholar

    [34]

    Pal A, Huse D A 2010 Phys. Rev. B 82 174411Google Scholar

    [35]

    Vosk R, Huse D A, Altman E 2015 Phys. Rev. X 5 031032

    [36]

    Agarwal K, Gopalakrishnan S, Knap M, Müller, Demler E 2015 Phys. Rev. Lett. 114 160401Google Scholar

    [37]

    Fan R, Zhang P, Shen H, Zhai H 2017 Sci. Bull. 62 707Google Scholar

    [38]

    Else D V, Bauer B, Nayak C 2016 Phys. Rev. Lett. 117 090402Google Scholar

    [39]

    Oganesyan V, Huse D A 2007 Phys. Rev. B 75 155111Google Scholar

    [40]

    Vosk R, Altman E 2013 Phys. Rev. Lett. 110 067204Google Scholar

    [41]

    Bauer B, Nayak C 2013 J. Stat. Mech. 2013 P09005Google Scholar

    [42]

    王玉成 2018 博士学位论文(北京: 中国科学院大学)

    Wang Y 2018 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese)

    [43]

    Serbyn M, Papić, Abanin D A 2013 Phys. Rev. Lett. 111 127201Google Scholar

    [44]

    Bardarson J H, Pollmann F, Moore J E 2012 Phys. Rev. Lett. 109 017202Google Scholar

    [45]

    Serbyn M, Papić Z, Abanin D A 2013 Phys. Rev. Lett. 110 260601Google Scholar

    [46]

    Bordia P, Lüschen H P, Scherg S, Gopalakrishnan S, Knap M, Schneider U, Bloch I 2017 Phys. Rev. X 7 041047

    [47]

    Kohlert T, Scherg S, Li X, Lüschen H P, Das Sarma S, Bloch I, Aidelsburger M 2018 arXiv:1809.04055

    [48]

    Hiramoto H, Abe S 1988 J. Phys. Soc. Jpn. 57 230Google Scholar

    [49]

    Hiramoto H, Abe S 1988 J. Phys. Soc. Jpn. 57 1365Google Scholar

    [50]

    Metzler R, Klafter J 2000 Phys. Rep. 339 1Google Scholar

    [51]

    Battiato M, Carva K, Oppeneer P M 2012 Phys. Rev. B 86 024404Google Scholar

    [52]

    Foster S, Thesberg M, Neophytou N 2017 Phys. Rev. B 96 195425Google Scholar

    [53]

    Ketzmerick R, Petschel G, Geisel T 1992 Phys. Rev. Lett. 69 695Google Scholar

    [54]

    Ketzmerick R, Kruse K, Kraut S, Geisel T 1997 Phys. Rev. Lett. 79 1959Google Scholar

    [55]

    Qin P, Yin C, Chen S 2014 Phys. Rev. B 90 054303

    [56]

    Floquet G 1883 Ann. ENS 12 47

    [57]

    Grifoni M, Hänggi P 1998 Phys. Rep. 304 229Google Scholar

    [58]

    Blümel R, Smilansky U 1984 Phys. Rev. Lett. 52 137Google Scholar

    [59]

    Izrailev F M 1988 Phys. Lett. A 134 13Google Scholar

    [60]

    Amico L, Fazio R, Osterloh A, Vedral V 2008 Rev. Mod. Phys. 80 517Google Scholar

    [61]

    Wilcox R M 1967 J. Math. Phys. 8 962Google Scholar

    [62]

    Heyl M, Polkovnikov A, Kehrein S 2013 Phys. Rev. Lett. 110 135704Google Scholar

    [63]

    Heyl M 2014 Phys. Rev. Lett. 113 205701Google Scholar

    [64]

    Heyl M 2015 Phys. Rev. Lett. 115 140602Google Scholar

    [65]

    Heyl M 2018 Rep. Prog. Phys. 81 054001Google Scholar

    [66]

    Yang C, Wang Y, Wang P, Gao X, Chen S 2017 Phys. Rev. B 95 184201Google Scholar

    [67]

    Jalabert R A, Pastawski H M 2001 Phys. Rev. Lett. 86 2490Google Scholar

    [68]

    Cucchietti F M, Dalvit D A R, Paz J P, Zurek W H 2003 Phys. Rev. Lett. 91 210403Google Scholar

    [69]

    Gorin T, Prosen T, Seligman T H, Znidaric M 2006 Phys. Rep. 435 33Google Scholar

    [70]

    Quan H T, Song Z, Liu X F, Zanardi P, Sun C P 2006 Phys. Rev. Lett. 96 140604Google Scholar

    [71]

    Jafari R, Johannesson H 2017 Phys. Rev. Lett. 118 015701Google Scholar

    [72]

    Yang B J, Nagaosa N 2014 Nat. Commun. 5 4898Google Scholar

    [73]

    Hassan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [74]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [75]

    Fang C, Weng H, Dai X, Fang Z 2016 Chin. Phys. B 25 117106Google Scholar

    [76]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698Google Scholar

    [77]

    Guo H, Chen S 2015 Phys. Rev. B 91 041402

    [78]

    Thouless D J 1983 Phys. Rev. B 27 6083Google Scholar

    [79]

    Wang L, Troyer M, Dai X 2013 Phys. Rev. Lett. 111 026802Google Scholar

    [80]

    Nakajima S, Tomita T, Taie S, Ichinose T, Ozawa H, Wang L, Troyer M, Takahashi Y 2016 Nat. Phys. 12 296Google Scholar

    [81]

    Xu Z, Zhang Y, Chen S 2017 Phys. Rev. A 96 013606Google Scholar

    [82]

    Kitaev A Y 2001 Phys. Usp. 44 131Google Scholar

    [83]

    Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101Google Scholar

    [84]

    Xu G, Weng H, Wang Z, Dai X, Fang Z 2011 Phys. Rev. Lett. 107 186806Google Scholar

    [85]

    Lv B Q, Xu N, Weng H M, Ma J Z, Richard P, Huang X C, Zhao L X, Chen G F, Matt C E, Bisti F, Strocov V N, Mesot J, Fang Z, Dai X, Qian T, Shiand M, Ding H 2015 Nat. Phys. 11 724Google Scholar

    [86]

    Alicea J 2012 Rep. Prog. Phys. 75 076501Google Scholar

    [87]

    Gangadharaiah S, Braunecker B, Simon P, Loss D 2011 Phys. Rev. Lett. 107 036801Google Scholar

    [88]

    朱国毅, 王瑞蕊, 张广铭 2017 物理 46 154Google Scholar

    Zhu G Y, Wang R R, Zhang G M 2017 Phys. 46 154Google Scholar

    [89]

    Lieb E, Schultz T, Mattis D 1961 Ann. Phys. 16 407Google Scholar

    [90]

    Dubček T, Kennedy C J, Lu L, Ketterle W, Soljačić M, Buljan H 2015 Phys. Rev. Lett. 114 225301Google Scholar

    [91]

    Wang Y, Chen S 2017 Phys. Rev. A 95 053634

    [92]

    Weiβe A, Wellein G, Alvermann A, Fehske H 2006 Rev. Mod. Phys. 78 275Google Scholar

    [93]

    Bera S, Sau J D, Roy B 2016 Phys. Rev. B 93 201302Google Scholar

    [94]

    Kobayashi K, Ohtsuki T, Imura K I, Herbut I F 2014 Phys. Rev. Lett. 112 016402Google Scholar

    [95]

    Pixley J H, Goswami P, Das Sarma S 2015 Phys. Rev. Lett. 115 076601Google Scholar

    [96]

    Pixley J H, Wilson J H, Huse D A, Gopalakrishnan S 2018 Phys. Rev. Lett. 120 207604Google Scholar

    [97]

    Yan Z, Bi R, Shen H, Lu L, Zhang S C, Wang Z 2017 Phys. Rev. B 96 041103Google Scholar

    [98]

    Wang Y, Hu H, Chen S 2018 Phys. Rev. B 98 205410Google Scholar

    [99]

    Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402Google Scholar

    [100]

    Lang L, Cai X, Chen S 2012 Phys. Rev. Lett. 108 220401Google Scholar

    [101]

    Madsen K, Bergholtz E J, Brouwer 2013 Phys. Rev. B 88 125118Google Scholar

  • 图 1  基态的倒参与率随$ \varDelta $的变化,这里固定$ J=1 $$ L=1000 $. 左右的插图分别展示了$ \varDelta=1.9 $$ \varDelta=2.1 $时系统的基态波函数的分布

    Fig. 1.  IPR of ground states as a function of $ \varDelta $ for this system with $ J=1 $ and $ L=1000 $. The left and right insets show the distribution of the ground state with $ \varDelta=1.9 $ and $ \varDelta=2.1 $ respectively.

    图 2  实验实现准周期晶格的原理示意图. J 描述的是主晶格最近邻格点之间的跃迁, $ 2\varDelta $是由次晶格导致的在位能最大的差别

    Fig. 2.  Sketch of the quasiperiodic lattice realized in the experiment. J describes the hopping between the nearest-neighbor sites of the primary lattice and $ 2\varDelta $ is the maximum shift of the on-site energy induced by the secondary lattice.

    图 3  实验原理图. 制备的初始CDW态,以及在局域、中间和扩展相中,经过一段时间演化后,分别对应的系统的末态 (a)初态分布,制备为CDW态(根据定义,有$ I > 0 $, $ \xi=0 $); (b)局域态($ I > 0 $, $ \xi=0 $);(c)中间态,对应于不同的能量存在局域态和扩展态($ I > 0 $, $ \xi>0 $); (d)扩展态($ I = 0 $, $ \xi>0 $)

    Fig. 3.  Schematics of the experiment. Schematic illustration of the initial CDW state and the states reached after time evolution in the localized, intermediate, and extended phase, respectively: (a) Initial state: CDW state ($ I > 0 $, $ \xi=0 $); (b) localized phase ($ I > 0 $, $ \xi=0 $); (c) the intermediate phase, extended and localized states coexist at different energies ($ I > 0 $, $ \xi>0 $); (d) extended phase ($ I = 0 $, $ \xi>0 $).

    图 4  (a)$ \langle r\rangle $$ h $的变化. 当系统尺寸为$ L=12 $$ L=14 $时用的样品数是$ 50 $, 当$ L=16 $时用的样品数是$ 30 $, 当$ L=18 $时用的样品数是$ 20 $; (b)平均的纠缠熵$ \langle S \rangle $$ {\rm d} \langle S \rangle / {\rm d}h $$ h $的变化. 当$ L=8 $$ L=10 $时用$ 500 $个样品, 当$ L=12 $时用$ 100 $个样品, 当$ L=14 $时用$ 30 $个样品. 相互作用强度始终被固定为$ U=0.4 $. 这里一个样品指的是任选一个初相位$ \theta $[21]

    Fig. 4.  (a) $ \langle r\rangle $ as a function of $ h $. Here we use 50 samples for $ L = 12 $ and $ L =14 $, 30 samples for $ L = 16 $, and 20 samples for $ L =18 $; (b) averaged entanglement entropy $ \langle S \rangle $ and $ {\rm d} \langle S \rangle / {\rm d}h $ versus $ h $. Here we use 500 samples for $ L = 8 $ and $ L = 10 $, 100 samples for $ L = 12 $ and 30 samples for $ L = 14 $. The interaction strength is fixed at $ U= 0.4 $. Here a sample is specified by choosing an initial phase$ \theta $[21].

    图 5  AA模型中取不同的$ \varDelta $$ \sqrt{\langle(\text{δ} x)^2\rangle} $随时间t的变化的对数-对数图, 这里固定$ \alpha=\frac{\sqrt{5}-1}{2} $, 跃迁强度$ J=1 $, 以及系统尺寸$ L=3000 $

    Fig. 5.  Log-log plot of the width $ \sqrt{\langle(\text{δ} x)^2\rangle} $ vs time t for several values of $ \varDelta $ in the AA model with $ \alpha=\frac{\sqrt{5}-1}{2} $, $ J=1 $ and $ L=3000 $.

    图 6  固定$ \lambda=1 $$ L=1500 $, 平均信息熵随周期$ T $的变化. 左上角的插图展示了平均纠缠熵的导数随周期$ T $的变化, 这里固定$ \lambda=0.8 $(蓝色), $ \lambda=1.2 $(红色), 和$ \lambda=1.6 $(绿色). 右下角的插图展示了平均纠缠熵随$ \lambda $的变化, 这里分别固定$ T=0.05 $(蓝色), $ T=0.3 $(红色), $ T=0.5 $(绿色)[55]

    Fig. 6.  The mean information entropy as a function of $ T $ for this system with $ \lambda=1 $ and $ L=1500 $. The left up inset shows the derivative of the mean information entropy as a function of $ T $ with fixed $ \lambda=0.8 $(blue), $ \lambda=1.2 $(red), and $ \lambda=1.6 $(green). The right down inset shows the derivative of the mean information entropy as a function of $ \lambda $ with $ T\!=\!0.05 $(blue), $ T\!=\!0.3 $(red), and $ T\!=\!0.5 $(green)[55].

    图 7  固定系统尺寸$ L=900 $, 平均信息熵随$ \lambda $$ T $的变化[55]

    Fig. 7.  The mean information entropy versus both $ \lambda $ and $ T $ for the system with $ L=900 $[55].

    图 8  $ \varDelta_{\rm f} $取不同值时Loschmidt echo的演化. 初态选准周期势强度为 $ \varDelta_{\rm i}\!=\!0.5 $((a), (b))和 $ \varDelta_{\rm i}\!=\!4 $((c), (d))的哈密顿量的基态[66]

    Fig. 8.  Evolution of Loschmidt echo in a long time with different $ \varDelta_{\rm f} $s. The initial state is chosen to be the ground state of the Hamiltonian with $ \varDelta_{\rm i}=0.5 $ ((a), (b))and $ \varDelta_{\rm i}=4 $((c), (d))[66].

    图 9  固定系统参数$ L=1000 $, $ T=6\times10^{5} $$ \varDelta_{\rm i}=0.5 $$ m (\varepsilon ) $$ \varDelta_{\rm f} $的变化: (a)不同的颜色对应不同的$ \varepsilon $值, 这里的初态是初始哈密顿量的基态; (b) 选取不同的初态, $ n $表示初始哈密顿量的第n个本征态. 在$ \varDelta_{\rm f}=2 $处, 可以清晰地看到一个相边界. 这里固定$ \varepsilon =0.01 $[66]

    Fig. 9.  The behavior of $ m $ versus $ \varDelta_{\rm f} $ for the system with $ L=1000 $, $ T=6\times10^{5} $ and $ \varDelta_{\rm i}=0.5 $: (a) Different colors correspond to different $ \varepsilon $s and the initial state is chosen to be the ground state of the initial Hamiltonian; (b) different choice of initial state with $ n $ standing for the $ n{\rm th} $ eigenstates of the initial Hamiltonian $ H(\varDelta_{\rm i}) $. A clear boundary can be seen at $ \varDelta_{\rm f}=2 $. Here we choose $ \varepsilon =0.01 $[66].

    图 10   (a) 固定两个p波配对强度$ \varDelta=0.5 $$ \varDelta=0.8 $时,MIPR随准周期势强度$ V $的变化,这里用的系统尺寸是$ L=1000 $; (b) 系统随p波配对强度$ \varDelta $和准周期势强度$ V $变化的相图,I:扩展相,II:临界相,III:局域相. 这里固定$ J=1 $

    Fig. 10.  (a) MIPR as a function of the incommensurate potential strength $ V $ at two p-wave pairing strength $ \varDelta=0.5 $ and $ \Delta=0.8 $. Here use $ L=1000 $; (b) phase diagram of this system with a p-wave pairing strength $ \varDelta $ and incommensurate potential strength $ V $. I: extended phase, II: critical phase and III: localized phase. Here fix $ J=1 $.

    图 11   (a) 在开边界条件下, 固定$ \varDelta=0.5 $$ L=500 $时系统的能谱; (b), (c)不同的$ V $值时最低激发模的$ \phi_i $((b))和$ \psi_i $((c))的分布[16]

    Fig. 11.  (a) Energy spectra of this system with $ \varDelta=0.5 $ and $ L=500 $ under OBC. The distributions of $ \phi_i $(b) and $ \psi_i $(c) for the lowest excitation with different $ V $[16].

    图 12   第N个本征态的IPR((a))和MIPR((b))随$ k_x $$ k_y $的变化, 这里固定$ V=1.9 $; 第N个本征态的IPR((c))和MIPR((d))随$ k_x $$ V $的变化, 这里固定$ k_y=\frac{\text{π}}{2} $; 第N个本征态的IPR((e))和MIPR((f))作为$ k_y $$ V $的函数, 这里固定$ k_x=0 $. 其他参数是$ L=300 $$ t_x=t_y=t_z=1 $[91]

    Fig. 12.  IPR((a))and MIPR((b)) as a function of $ k_x $ and $ k_y $ with fixed $ V=1.9 $; IPR((c)) and MIPR((d)) as a function of $ k_x $ and $ V $ with fixed $ k_y=\frac{\text{π}}{2} $; IPR((e)) and MIPR((f)) as a function of $ k_y $ and $ V $ with fixed $ k_x=0 $. The lattice size is $ L=300 $ and $ t_x=t_y=t_z=1 $[91].

    图 13   (a) 不同晶格尺寸$ N $时, $ \rho(0) $$ V $的变化, 这里固定$ \sigma=0.02 $; (b)固定$ N=300 $, 取不同的准周期势强度$ V $时系统的态密度随能量的变化[91]

    Fig. 13.  (a) $ \rho(0) $ versus $ V $ for different lattice size $ N $ with fixed $ \sigma=0.02 $; (b) DOS with $ N=300 $ as a function of energy for various values of incommensurate potential strength $ V $[91].

  • [1]

    王义遒 2007 原子的激光冷却与陷俘(北京: 北京大学出版社)

    Wang Y Q 2007 Atomic Laser Cooling And Trapping (Beijing: Peking University Press) (in Chinese)

    [2]

    Stwalley W C 1976 Phys. Rev. Lett. 37 1628Google Scholar

    [3]

    Bloch I, Dalibard J, Zwerger W 2008 Rev. Mod. Phys. 80 885Google Scholar

    [4]

    Lin Y J, Jimenez-Garcia K, Spielman I B 2011 Nature 471 83Google Scholar

    [5]

    Wu Z, Zhang L, Sun W, Xu X T, Wang B Z, Ji S C, Deng Y J, Chen S, Liu X J, Pan J W 2016 Science 354 83Google Scholar

    [6]

    Zhang L, Liu X J 2018 arXiv:1806.05628

    [7]

    Cooper N R, Dalibard J, Spielman I B 2018 arXiv:1803.00249

    [8]

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

    [9]

    Hu J, Feng L, Zhang Z, Chin C, 2018 arXiv:1807.07504

    [10]

    Aubry S, André G 1980 Ann. Israel Phys. Soc 3 18

    [11]

    Das Sarma S, He S, Xie X C 1988 Phys. Rev. Lett. 61 2144Google Scholar

    [12]

    Das Sarma S, He S, Xie X C 1990 Phys. Rev. B 41 5544Google Scholar

    [13]

    Biddle J, Das Sarma S 2010 Phys. Rev. Lett. 104 070601Google Scholar

    [14]

    Biddle J, Priour Jr D J, Wang B, Das Sarma S 2011 Phys. Rev. B 83 075105Google Scholar

    [15]

    Roati G, D’Errico C, Fallani L, Fattori M, Fort C, Zaccanti M, Modugno G, Modugno M, Inguscio M 2008 Nature 453 895Google Scholar

    [16]

    Cai X, Lang L J, Chen S, Wang Y 2013 Phys. Rev. Lett. 110 176403Google Scholar

    [17]

    DeGottardi W, Sen D, Vishveshwara S 2013 Phys. Rev. Lett. 110 146404Google Scholar

    [18]

    Wang J, Liu X J, Gao X, Hu H 2016 Phys. Rev. B 93 104504Google Scholar

    [19]

    Wang Y, Wang Y, Chen S 2016 Eur. Phys. J. B 89 254Google Scholar

    [20]

    Iyer S, Oganesyan V, Refael G, Huse D A 2013 Phys. Rev. B 87 134202Google Scholar

    [21]

    Wang Y, Hu H, Chen S 2016 Eur. Phys. J. B 89 77Google Scholar

    [22]

    Setiawan F, Deng D L, Pixley J H 2017 Phys. Rev. B 96 104205Google Scholar

    [23]

    Schreiber M, Hodgman S S, Bordia P, Lüschen H P, Fischer M H, Vosk R, Altman E, Schneider U, Bloch I 2015 Science 349 842Google Scholar

    [24]

    Anderson P W 1958 Phys. Rev. 109 1492Google Scholar

    [25]

    Thouless D J 1974 Phys. Rep. 13 93Google Scholar

    [26]

    Schreiber M 1985 J. Phys. C 18 2493Google Scholar

    [27]

    Hashimoto Y, Niizeki K, Okabe Y 1992 J. Phys. A 25 5211Google Scholar

    [28]

    Modugno M 2009 New J. Phys. 11 033023Google Scholar

    [29]

    郎利君 2014 博士学位论文(北京: 中国科学院大学)

    Lang L J 2014 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese)

    [30]

    Li X, Li X, Das Sarma S 2017 Phys. Rev. B 96 085119Google Scholar

    [31]

    Lüschen H P, Scherg S, Kohlert T, Schreiber M, Bordia P, Li X, Das Sarma S, Bloch I 2018 Phys. Rev. Lett. 120 160404Google Scholar

    [32]

    Basko D, Aleiner I, Altshuler B 2006 Ann. Phys. 321 1126Google Scholar

    [33]

    Nandkishore R, Huse D A 2015 Annu. Rev. Condens. Matter Phys. 6 15Google Scholar

    [34]

    Pal A, Huse D A 2010 Phys. Rev. B 82 174411Google Scholar

    [35]

    Vosk R, Huse D A, Altman E 2015 Phys. Rev. X 5 031032

    [36]

    Agarwal K, Gopalakrishnan S, Knap M, Müller, Demler E 2015 Phys. Rev. Lett. 114 160401Google Scholar

    [37]

    Fan R, Zhang P, Shen H, Zhai H 2017 Sci. Bull. 62 707Google Scholar

    [38]

    Else D V, Bauer B, Nayak C 2016 Phys. Rev. Lett. 117 090402Google Scholar

    [39]

    Oganesyan V, Huse D A 2007 Phys. Rev. B 75 155111Google Scholar

    [40]

    Vosk R, Altman E 2013 Phys. Rev. Lett. 110 067204Google Scholar

    [41]

    Bauer B, Nayak C 2013 J. Stat. Mech. 2013 P09005Google Scholar

    [42]

    王玉成 2018 博士学位论文(北京: 中国科学院大学)

    Wang Y 2018 Ph. D. Dissertation (Beijing: University of Chinese Academy of Sciences) (in Chinese)

    [43]

    Serbyn M, Papić, Abanin D A 2013 Phys. Rev. Lett. 111 127201Google Scholar

    [44]

    Bardarson J H, Pollmann F, Moore J E 2012 Phys. Rev. Lett. 109 017202Google Scholar

    [45]

    Serbyn M, Papić Z, Abanin D A 2013 Phys. Rev. Lett. 110 260601Google Scholar

    [46]

    Bordia P, Lüschen H P, Scherg S, Gopalakrishnan S, Knap M, Schneider U, Bloch I 2017 Phys. Rev. X 7 041047

    [47]

    Kohlert T, Scherg S, Li X, Lüschen H P, Das Sarma S, Bloch I, Aidelsburger M 2018 arXiv:1809.04055

    [48]

    Hiramoto H, Abe S 1988 J. Phys. Soc. Jpn. 57 230Google Scholar

    [49]

    Hiramoto H, Abe S 1988 J. Phys. Soc. Jpn. 57 1365Google Scholar

    [50]

    Metzler R, Klafter J 2000 Phys. Rep. 339 1Google Scholar

    [51]

    Battiato M, Carva K, Oppeneer P M 2012 Phys. Rev. B 86 024404Google Scholar

    [52]

    Foster S, Thesberg M, Neophytou N 2017 Phys. Rev. B 96 195425Google Scholar

    [53]

    Ketzmerick R, Petschel G, Geisel T 1992 Phys. Rev. Lett. 69 695Google Scholar

    [54]

    Ketzmerick R, Kruse K, Kraut S, Geisel T 1997 Phys. Rev. Lett. 79 1959Google Scholar

    [55]

    Qin P, Yin C, Chen S 2014 Phys. Rev. B 90 054303

    [56]

    Floquet G 1883 Ann. ENS 12 47

    [57]

    Grifoni M, Hänggi P 1998 Phys. Rep. 304 229Google Scholar

    [58]

    Blümel R, Smilansky U 1984 Phys. Rev. Lett. 52 137Google Scholar

    [59]

    Izrailev F M 1988 Phys. Lett. A 134 13Google Scholar

    [60]

    Amico L, Fazio R, Osterloh A, Vedral V 2008 Rev. Mod. Phys. 80 517Google Scholar

    [61]

    Wilcox R M 1967 J. Math. Phys. 8 962Google Scholar

    [62]

    Heyl M, Polkovnikov A, Kehrein S 2013 Phys. Rev. Lett. 110 135704Google Scholar

    [63]

    Heyl M 2014 Phys. Rev. Lett. 113 205701Google Scholar

    [64]

    Heyl M 2015 Phys. Rev. Lett. 115 140602Google Scholar

    [65]

    Heyl M 2018 Rep. Prog. Phys. 81 054001Google Scholar

    [66]

    Yang C, Wang Y, Wang P, Gao X, Chen S 2017 Phys. Rev. B 95 184201Google Scholar

    [67]

    Jalabert R A, Pastawski H M 2001 Phys. Rev. Lett. 86 2490Google Scholar

    [68]

    Cucchietti F M, Dalvit D A R, Paz J P, Zurek W H 2003 Phys. Rev. Lett. 91 210403Google Scholar

    [69]

    Gorin T, Prosen T, Seligman T H, Znidaric M 2006 Phys. Rep. 435 33Google Scholar

    [70]

    Quan H T, Song Z, Liu X F, Zanardi P, Sun C P 2006 Phys. Rev. Lett. 96 140604Google Scholar

    [71]

    Jafari R, Johannesson H 2017 Phys. Rev. Lett. 118 015701Google Scholar

    [72]

    Yang B J, Nagaosa N 2014 Nat. Commun. 5 4898Google Scholar

    [73]

    Hassan M Z, Kane C L 2010 Rev. Mod. Phys. 82 3045Google Scholar

    [74]

    Qi X L, Zhang S C 2011 Rev. Mod. Phys. 83 1057Google Scholar

    [75]

    Fang C, Weng H, Dai X, Fang Z 2016 Chin. Phys. B 25 117106Google Scholar

    [76]

    Su W P, Schrieffer J R, Heeger A J 1979 Phys. Rev. Lett. 42 1698Google Scholar

    [77]

    Guo H, Chen S 2015 Phys. Rev. B 91 041402

    [78]

    Thouless D J 1983 Phys. Rev. B 27 6083Google Scholar

    [79]

    Wang L, Troyer M, Dai X 2013 Phys. Rev. Lett. 111 026802Google Scholar

    [80]

    Nakajima S, Tomita T, Taie S, Ichinose T, Ozawa H, Wang L, Troyer M, Takahashi Y 2016 Nat. Phys. 12 296Google Scholar

    [81]

    Xu Z, Zhang Y, Chen S 2017 Phys. Rev. A 96 013606Google Scholar

    [82]

    Kitaev A Y 2001 Phys. Usp. 44 131Google Scholar

    [83]

    Wan X, Turner A M, Vishwanath A, Savrasov S Y 2011 Phys. Rev. B 83 205101Google Scholar

    [84]

    Xu G, Weng H, Wang Z, Dai X, Fang Z 2011 Phys. Rev. Lett. 107 186806Google Scholar

    [85]

    Lv B Q, Xu N, Weng H M, Ma J Z, Richard P, Huang X C, Zhao L X, Chen G F, Matt C E, Bisti F, Strocov V N, Mesot J, Fang Z, Dai X, Qian T, Shiand M, Ding H 2015 Nat. Phys. 11 724Google Scholar

    [86]

    Alicea J 2012 Rep. Prog. Phys. 75 076501Google Scholar

    [87]

    Gangadharaiah S, Braunecker B, Simon P, Loss D 2011 Phys. Rev. Lett. 107 036801Google Scholar

    [88]

    朱国毅, 王瑞蕊, 张广铭 2017 物理 46 154Google Scholar

    Zhu G Y, Wang R R, Zhang G M 2017 Phys. 46 154Google Scholar

    [89]

    Lieb E, Schultz T, Mattis D 1961 Ann. Phys. 16 407Google Scholar

    [90]

    Dubček T, Kennedy C J, Lu L, Ketterle W, Soljačić M, Buljan H 2015 Phys. Rev. Lett. 114 225301Google Scholar

    [91]

    Wang Y, Chen S 2017 Phys. Rev. A 95 053634

    [92]

    Weiβe A, Wellein G, Alvermann A, Fehske H 2006 Rev. Mod. Phys. 78 275Google Scholar

    [93]

    Bera S, Sau J D, Roy B 2016 Phys. Rev. B 93 201302Google Scholar

    [94]

    Kobayashi K, Ohtsuki T, Imura K I, Herbut I F 2014 Phys. Rev. Lett. 112 016402Google Scholar

    [95]

    Pixley J H, Goswami P, Das Sarma S 2015 Phys. Rev. Lett. 115 076601Google Scholar

    [96]

    Pixley J H, Wilson J H, Huse D A, Gopalakrishnan S 2018 Phys. Rev. Lett. 120 207604Google Scholar

    [97]

    Yan Z, Bi R, Shen H, Lu L, Zhang S C, Wang Z 2017 Phys. Rev. B 96 041103Google Scholar

    [98]

    Wang Y, Hu H, Chen S 2018 Phys. Rev. B 98 205410Google Scholar

    [99]

    Kraus Y E, Lahini Y, Ringel Z, Verbin M, Zilberberg O 2012 Phys. Rev. Lett. 109 106402Google Scholar

    [100]

    Lang L, Cai X, Chen S 2012 Phys. Rev. Lett. 108 220401Google Scholar

    [101]

    Madsen K, Bergholtz E J, Brouwer 2013 Phys. Rev. B 88 125118Google Scholar

  • [1] 李婷, 汪涛, 王叶兵, 卢本全, 卢晓同, 尹默娟, 常宏. 浅光晶格中量子隧穿现象的实验观测. 物理学报, 2022, 71(7): 073701. doi: 10.7498/aps.71.20212038
    [2] 姚杰, 赵爱迪. 表面单分子量子态的探测和调控研究进展. 物理学报, 2022, 71(6): 060701. doi: 10.7498/aps.71.20212324
    [3] 张志强. 简谐与光晶格复合势阱中旋转二维玻色-爱因斯坦凝聚体中的涡旋链. 物理学报, 2022, 71(22): 220304. doi: 10.7498/aps.71.20221312
    [4] 张爱霞, 姜艳芳, 薛具奎. 光晶格中自旋轨道耦合玻色-爱因斯坦凝聚体的非线性能谱特性. 物理学报, 2021, 70(20): 200302. doi: 10.7498/aps.70.20210705
    [5] 姜天舒, 肖孟, 张昭庆, 陈子亭. 周期与非周期传输线网络的物理与拓扑性质. 物理学报, 2020, 69(15): 150301. doi: 10.7498/aps.69.20200258
    [6] 文凯, 王良伟, 周方, 陈良超, 王鹏军, 孟增明, 张靖. 超冷87Rb原子在二维光晶格中Mott绝缘态的实验实现. 物理学报, 2020, 69(19): 193201. doi: 10.7498/aps.69.20200513
    [7] 卢晓同, 李婷, 孔德欢, 王叶兵, 常宏. 锶原子光晶格钟碰撞频移的测量. 物理学报, 2019, 68(23): 233401. doi: 10.7498/aps.68.20191147
    [8] 赵兴东, 张莹莹, 刘伍明. 光晶格中超冷原子系统的磁激发. 物理学报, 2019, 68(4): 043703. doi: 10.7498/aps.68.20190153
    [9] 徐志浩, 皇甫宏丽, 张云波. 一维准周期晶格中玻色子对的迁移率边. 物理学报, 2019, 68(8): 087201. doi: 10.7498/aps.68.20182218
    [10] 李晓云, 孙博文, 许正倩, 陈静, 尹亚玲, 印建平. 基于调制光晶格的中性分子束光学Stark减速与囚禁的理论研究. 物理学报, 2018, 67(20): 203702. doi: 10.7498/aps.67.20181348
    [11] 林弋戈, 方占军. 锶原子光晶格钟. 物理学报, 2018, 67(16): 160604. doi: 10.7498/aps.67.20181097
    [12] 陈海军. 变分法研究二维光晶格中玻色-爱因斯坦凝聚的调制不稳定性. 物理学报, 2015, 64(5): 054702. doi: 10.7498/aps.64.054702
    [13] 田晓, 王叶兵, 卢本全, 刘辉, 徐琴芳, 任洁, 尹默娟, 孔德欢, 常宏, 张首刚. 锶玻色子的“魔术”波长光晶格装载实验研究. 物理学报, 2015, 64(13): 130601. doi: 10.7498/aps.64.130601
    [14] 藤斐, 谢征微. 光晶格中双组分玻色-爱因斯坦凝聚系统的调制不稳定性. 物理学报, 2013, 62(2): 026701. doi: 10.7498/aps.62.026701
    [15] 何龙, 宋筠. 双层石墨烯材料中无序导致超导-绝缘体相变的数值研究. 物理学报, 2013, 62(5): 057303. doi: 10.7498/aps.62.057303
    [16] 张科智, 王建军, 刘国荣, 薛具奎. 两组分BECs在光晶格中的隧穿动力学及其周期调制效应. 物理学报, 2010, 59(5): 2952-2961. doi: 10.7498/aps.59.2952
    [17] 赵义. 一维长程关联无序系统的局域性. 物理学报, 2010, 59(1): 532-535. doi: 10.7498/aps.59.532
    [18] 周骏, 任海东, 冯亚萍. 强非局域光晶格中空间孤子的脉动传播. 物理学报, 2010, 59(6): 3992-4000. doi: 10.7498/aps.59.3992
    [19] 黄劲松, 陈海峰, 谢征微. 光晶格中双组分偶极玻色-爱因斯坦凝聚体的调制不稳定性. 物理学报, 2008, 57(6): 3435-3439. doi: 10.7498/aps.57.3435
    [20] 徐志君, 程 成, 杨欢耸, 武 强, 熊宏伟. 三维光晶格中玻色凝聚气体基态波函数及干涉演化. 物理学报, 2004, 53(9): 2835-2842. doi: 10.7498/aps.53.2835
计量
  • 文章访问数:  13186
  • PDF下载量:  423
  • 被引次数: 0
出版历程
  • 收稿日期:  2018-10-30
  • 修回日期:  2018-12-25
  • 上网日期:  2019-02-01
  • 刊出日期:  2019-02-20

/

返回文章
返回