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Simulating Dirac, Weyl and Maxwell equations with cold atoms in optical lattices

Zhu Yan-Qing Zhang Dan-Wei Zhu Shi-Liang

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Simulating Dirac, Weyl and Maxwell equations with cold atoms in optical lattices

Zhu Yan-Qing, Zhang Dan-Wei, Zhu Shi-Liang
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  • Relativistic wave equations, such as Dirac, Weyl or Maxwell equations, are fundamental equations which we use to describe the dynamics of the microscopic particles. On the other hand, recent experimental and theoretical studies have shown that almost all parameters in cold atomic systems are precisely tunable, so the cold atom systems are considered as an ideal platform to perform quantum simulations. It can be used to study some topics in high energy and condensed matter physics. In this article, we will first introduce the ideas and methods for engineering the Hamiltonian of atoms, mainly related to the theories of laser-assisted tunneling. Based on these methods, one can simulate the equations of motion of relativistic particles and observe some interesting behaviors which are hard to be observed in other systems. The article reviews these recent advances.
      Corresponding author: Zhang Dan-Wei, zdanwei@126.com ; Zhu Shi-Liang, slzhu@nju.edu.cn
    • Funds: Project supported by the National Key Research and Development Program of China (Grant No. 2016YFA0301803), the National Natural Science Foundation of China (Grant Nos. 11604103, 91636218, 11474153), and the Natural Science Foundation of Guangdong Province, China (Grant No. 2016A030313436).
    [1]

    Chu S 1998 Rev. Mod. Phys. 70 685Google Scholar

    [2]

    Cohen-Tannoudji C N 1998 Rev. Mod. Phys. 70 707Google Scholar

    [3]

    Phillips W D 1998 Rev. Mod. Phys. 70 721Google Scholar

    [4]

    Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198Google Scholar

    [5]

    Davis K B, Mewes M O, Andrews M R, van Druten N J, Durfee D S, Kurn D M, Ketterle W 1995 Phys. Rev. Lett. 75 3969Google Scholar

    [6]

    Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225Google Scholar

    [7]

    Jessen P, Deutsch I 1996 Adv. At. Mol. Opt. Phys. 37 95Google Scholar

    [8]

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

    [9]

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

    [10]

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

    [11]

    Zhang D W, Zhu Y Q, Zhao Y X, Hui Y, Zhu S L 2018 arXiv: 1810.09228

    [12]

    Zhu S L, Zhang D W, Wang Z D 2009 Phys. Rev. Lett. 102 210403Google Scholar

    [13]

    Lewenstein M, Sanpera A, Ahufinger V, Damski B, Sen A, Sen U 2007 Adv. Phys. 56 243Google Scholar

    [14]

    Jaksch D, Zoller P 2003 New J. Phys. 5 56Google Scholar

    [15]

    Gerbier F, Dalibard J 2010 New J. Phys. 12 033007Google Scholar

    [16]

    Struck J, Olschlager C, Weinberg M, et al. 2012 Phys. Rev. Lett. 108 225304Google Scholar

    [17]

    Grimm R, Weidemüller M 2000 Adv. At. Mol. Opt. Phys. 42 95Google Scholar

    [18]

    Zhu S L, Wang B, Duan L M 2007 Phys. Rev. Lett. 98 260402Google Scholar

    [19]

    Zhang D W, Shan C J, Mei F, Yang M, Wang R Q, Zhu S L 2014 Phys. Rev. A 89 015601Google Scholar

    [20]

    Mandel O, Greiner M, Widera A, Rom T, Hansch T W, Bloch I 2003 Phys. Rev. Lett. 91 010407Google Scholar

    [21]

    Lee P J, Anderlini M, Brown B L, Sebby-Strabley J, Phillips W D, Porto J V 2007 Phys. Rev. Lett. 99 020402Google Scholar

    [22]

    Mazza L, Bermudez A, Goldman N, Rizzi M, Martin-Delgado M A, Lewenstein M 2012 New J. Phys. 14 015007Google Scholar

    [23]

    Aidelsburger M, Atala M, Nascimbène M, Trotzky S, Chen Y A, Bloch I 2011 Phys. Rev. Lett. 107 255301Google Scholar

    [24]

    Aidelsburger M, Atala M, Lohse M, Barreiro J T, Paredes B, Bloch I 2013 Phys. Rev. Lett. 111 185301Google Scholar

    [25]

    Miyake H, Siviloglou G A, Kennedy C J, Burton W C, Ketterle W 2013 Phys. Rev. Lett. 111 185302Google Scholar

    [26]

    Tarruell L, Greif D, Uehlinger T, Jotzu G, Esslinger T 2012 Nature 483 302Google Scholar

    [27]

    Lim L K, Fuchs J N, Montambaux G 2012 Phys. Rev. Lett. 108 175303Google Scholar

    [28]

    Uehlinger T, Greif D, Jotzu G, Tarruell L, Esslinger T, Wang L, Troyer M 2013 Eur. Phys. J. Special Topics 217 121Google Scholar

    [29]

    Duca L, Li T, Reitter M, Bloch I, Schleier-Smith M, Schneider U 2015 Science 347 288Google Scholar

    [30]

    Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001Google Scholar

    [31]

    Bermudez A, Mazza L, Rizzi M, Goldman N, Lewenstein M, Martin-Delgado M A 2010 Phys. Rev. Lett. 105 190404Google Scholar

    [32]

    Mazza L, Bermudez A, Goldman N, Rizzi M, Martin-Delgado M A, Lewenstein M 2012 New J. Phys. 14 015007

    [33]

    Yang M, Zhu S L 2010 Phys. Rev. A 82 064102Google Scholar

    [34]

    Lepori L, Mussardo G, Trombettoni A 2010 Europhys. Lett. 92 50003Google Scholar

    [35]

    Wilson K, New Phenomena in Subnuclear Physics, Plenum, New York, 1977.

    [36]

    Zhang D W, Mei F, Xue Z Y, Zhu S L, Wang Z D 2015 Phys. Rev. A 92 013612Google Scholar

    [37]

    Ganeshan S, Sarma S D 2015 Phys. Rev. B 91 125438Google Scholar

    [38]

    Jiang J H 2012 Phys. Rev. A 85 033640Google Scholar

    [39]

    He W Y, Zhang S, Law K T 2016 Phys. Rev. A 94 013606Google Scholar

    [40]

    Hou J M, Chen W 2016 Sci. Rep. 6 33512Google Scholar

    [41]

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

    [42]

    Xu Y, Duan L M 2016 Phys. Rev. A 94 053619Google Scholar

    [43]

    Shastri K, Yang Z, Zhang B 2017 Phys. Rev. B 95 014306Google Scholar

    [44]

    Kong X, He J, Liang Y, Kou S 2017 Phys. Rev. A 95 33629Google Scholar

    [45]

    Zhu Y Q, Zhang D W, Yan H, Xing D Y, Zhu S L 2017 Phys. Rev. A 96 033634Google Scholar

    [46]

    Tan X, Zhang D W, Liu Q, Xue G, Yu H F, Zhu Y Q, Yan H, Zhu S L, Yu Y 2018 Phys. Rev. Lett. 120 130503Google Scholar

    [47]

    Liang L, Yu Y 2016 Phys. Rev. B 93 045113Google Scholar

    [48]

    Lan Z, Goldman N, Bermudez A, Lu W, Öhberg P 2011 Phys. Rev. B 84 165115Google Scholar

    [49]

    Kitaev A, Laumann C 2009 arXiv: 0904.2771

    [50]

    Trebst S, Troyer M, Wang Z, Ludwig A W W 2008 Prog. Theor. Phys. Supp. 176 384Google Scholar

    [51]

    Nayak C, Simon S H, Stern A, Freedman M, Sarma S D 2008 Rev. Mod. Phys. 80 1083Google Scholar

    [52]

    Read N, Rezayi E 1999 Phys. Rev. B 59 8084Google Scholar

    [53]

    Liu S, Shan C J, Zhang Z M, Xue Z Y 2014 Quantum Inf. Process. 13 1813Google Scholar

    [54]

    Vaishnav J Y, Clark C W 2008 Phys. Rev. Lett. 100 153002Google Scholar

    [55]

    Zhang D W, Xue Z Y, Yan H, Wang Z D, Zhu S L 2012 Phys. Rev. A 85 013628Google Scholar

    [56]

    Li Z, Wang H Q, Zhang D W, Zhu S L, Xing D Y 2016 Phys. Rev. A 94 043617Google Scholar

    [57]

    Xu Y, Duan L M 2017 Phys. Rev. B 96 155301Google Scholar

    [58]

    Shen X, Zhu Y Q, Li Z (In preparation)

    [59]

    Bliokh K Y, Smirnova D, Nori F 2015 Science 348 1448Google Scholar

    [60]

    邱英, 何军, 王彦华, 王婧, 张天才, 王军民 2008 物理学报 57 6227Google Scholar

    Qiu Y, He J, Wang Y H, Wang J, Zhang T C, Wang J M 2008 Acta Phys. Sin. 57 6227Google Scholar

    [61]

    Atala M, Aidelsburger M, Barreiro J T, Abanin D, Kitagawa T, Demler E, Bloch I 2013 Nat. Phys. 9 795Google Scholar

    [62]

    Fisher M P A, Weichwan P B, Grinstein G, Fisher D S 1989 Phys. Rev. B 40 546Google Scholar

    [63]

    Jaksch D, Bruder C, Cirac J I, Zoller P 1998 Phys. Rev. Lett. 81 3108Google Scholar

    [64]

    Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T 2014 Nature 515 237Google Scholar

    [65]

    Aidelsburger M, Lohse M, Schweizer C, Atala M, Barreiro J T, Nascimbène S, Cooper N R, Bloch I, Goldman N 2015 Nat. Phys. 11 162Google Scholar

    [66]

    杨圆, 陈帅, 李小兵 2018 物理学报 67 237101Google Scholar

    Yang Y, Chen S, Li X B 2018 Acta Phys. Sin. 67 237101Google Scholar

    [67]

    范桁 2018 物理学报 67 120301Google Scholar

    Fan H 2018 Acta Phys. Sin. 67 120301Google Scholar

  • 图 1  基于激光辅助跳跃实现人工磁场, 黑(灰)色圆分别表示内态为$ |g\rangle $$ (|e\rangle) $的Yb原子 (a)内态被标记为$ |g\rangle $$ |e\rangle $的原子被囚禁在自旋依赖的光晶格势$ V_g $$ V_e $中, 其中$ V_g=-V_e $; (b) $ x $方向上的激光辅助跃迁; (c)自旋依赖光晶格示意图. $ y $方向存在自然跳跃, $ x $方向由一束拉曼光$ \varOmega_{\rm R} $诱导跳跃

    Figure 1.  Realization of artificial magnetic field based on laser-assisted tunneling. Gray and black dots represent the Yb atoms correspond to internal states $|g\rangle$ and $|e\rangle$, respectively: (a) The atoms $|g\rangle$ and $|e\rangle$ are trapped in the state-dependent optical lattice potentials $V_g$ and $V_e$, where $V_g=-V_e$; (b) laser-assisted tunneling along $x$ direction; (c) sketch of state-dependent optical lattice. Nature tunneling occurs along the $y$ direction, and the tunneling along $x$ direction is induced by a Raman beam $\varOmega_{\rm R}$.

    图 2  (a)交错磁通光晶格; (b)双光子拉曼过程; (c)等效${\text{π}}$磁通

    Figure 2.  (a) Staggered flux optical lattice; (b) two-photon Raman process; (c) effective ${\text{π}}$ flux.

    图 3  实现外尔半金属的三维立方晶格示意图. 合理设计$x$$z$方向跳跃, 在动量空间会出现外尔点. 虚线和实线分别表示获得相位${\text{π}}$和0[41]

    Figure 3.  Schematic diagram of a three-dimensional cubic lattice of a Weyl semimetal. The Weyl points will be created in the momentum space if the tunneling along $x$ and $z$ directions are well-designed . The dashed and solid lines indicate the phase ${\text{π}}$ and 0, respectively.

  • [1]

    Chu S 1998 Rev. Mod. Phys. 70 685Google Scholar

    [2]

    Cohen-Tannoudji C N 1998 Rev. Mod. Phys. 70 707Google Scholar

    [3]

    Phillips W D 1998 Rev. Mod. Phys. 70 721Google Scholar

    [4]

    Anderson M H, Ensher J R, Matthews M R, Wieman C E, Cornell E A 1995 Science 269 198Google Scholar

    [5]

    Davis K B, Mewes M O, Andrews M R, van Druten N J, Durfee D S, Kurn D M, Ketterle W 1995 Phys. Rev. Lett. 75 3969Google Scholar

    [6]

    Chin C, Grimm R, Julienne P, Tiesinga E 2010 Rev. Mod. Phys. 82 1225Google Scholar

    [7]

    Jessen P, Deutsch I 1996 Adv. At. Mol. Opt. Phys. 37 95Google Scholar

    [8]

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

    [9]

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

    [10]

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

    [11]

    Zhang D W, Zhu Y Q, Zhao Y X, Hui Y, Zhu S L 2018 arXiv: 1810.09228

    [12]

    Zhu S L, Zhang D W, Wang Z D 2009 Phys. Rev. Lett. 102 210403Google Scholar

    [13]

    Lewenstein M, Sanpera A, Ahufinger V, Damski B, Sen A, Sen U 2007 Adv. Phys. 56 243Google Scholar

    [14]

    Jaksch D, Zoller P 2003 New J. Phys. 5 56Google Scholar

    [15]

    Gerbier F, Dalibard J 2010 New J. Phys. 12 033007Google Scholar

    [16]

    Struck J, Olschlager C, Weinberg M, et al. 2012 Phys. Rev. Lett. 108 225304Google Scholar

    [17]

    Grimm R, Weidemüller M 2000 Adv. At. Mol. Opt. Phys. 42 95Google Scholar

    [18]

    Zhu S L, Wang B, Duan L M 2007 Phys. Rev. Lett. 98 260402Google Scholar

    [19]

    Zhang D W, Shan C J, Mei F, Yang M, Wang R Q, Zhu S L 2014 Phys. Rev. A 89 015601Google Scholar

    [20]

    Mandel O, Greiner M, Widera A, Rom T, Hansch T W, Bloch I 2003 Phys. Rev. Lett. 91 010407Google Scholar

    [21]

    Lee P J, Anderlini M, Brown B L, Sebby-Strabley J, Phillips W D, Porto J V 2007 Phys. Rev. Lett. 99 020402Google Scholar

    [22]

    Mazza L, Bermudez A, Goldman N, Rizzi M, Martin-Delgado M A, Lewenstein M 2012 New J. Phys. 14 015007Google Scholar

    [23]

    Aidelsburger M, Atala M, Nascimbène M, Trotzky S, Chen Y A, Bloch I 2011 Phys. Rev. Lett. 107 255301Google Scholar

    [24]

    Aidelsburger M, Atala M, Lohse M, Barreiro J T, Paredes B, Bloch I 2013 Phys. Rev. Lett. 111 185301Google Scholar

    [25]

    Miyake H, Siviloglou G A, Kennedy C J, Burton W C, Ketterle W 2013 Phys. Rev. Lett. 111 185302Google Scholar

    [26]

    Tarruell L, Greif D, Uehlinger T, Jotzu G, Esslinger T 2012 Nature 483 302Google Scholar

    [27]

    Lim L K, Fuchs J N, Montambaux G 2012 Phys. Rev. Lett. 108 175303Google Scholar

    [28]

    Uehlinger T, Greif D, Jotzu G, Tarruell L, Esslinger T, Wang L, Troyer M 2013 Eur. Phys. J. Special Topics 217 121Google Scholar

    [29]

    Duca L, Li T, Reitter M, Bloch I, Schleier-Smith M, Schneider U 2015 Science 347 288Google Scholar

    [30]

    Armitage N P, Mele E J, Vishwanath A 2018 Rev. Mod. Phys. 90 015001Google Scholar

    [31]

    Bermudez A, Mazza L, Rizzi M, Goldman N, Lewenstein M, Martin-Delgado M A 2010 Phys. Rev. Lett. 105 190404Google Scholar

    [32]

    Mazza L, Bermudez A, Goldman N, Rizzi M, Martin-Delgado M A, Lewenstein M 2012 New J. Phys. 14 015007

    [33]

    Yang M, Zhu S L 2010 Phys. Rev. A 82 064102Google Scholar

    [34]

    Lepori L, Mussardo G, Trombettoni A 2010 Europhys. Lett. 92 50003Google Scholar

    [35]

    Wilson K, New Phenomena in Subnuclear Physics, Plenum, New York, 1977.

    [36]

    Zhang D W, Mei F, Xue Z Y, Zhu S L, Wang Z D 2015 Phys. Rev. A 92 013612Google Scholar

    [37]

    Ganeshan S, Sarma S D 2015 Phys. Rev. B 91 125438Google Scholar

    [38]

    Jiang J H 2012 Phys. Rev. A 85 033640Google Scholar

    [39]

    He W Y, Zhang S, Law K T 2016 Phys. Rev. A 94 013606Google Scholar

    [40]

    Hou J M, Chen W 2016 Sci. Rep. 6 33512Google Scholar

    [41]

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

    [42]

    Xu Y, Duan L M 2016 Phys. Rev. A 94 053619Google Scholar

    [43]

    Shastri K, Yang Z, Zhang B 2017 Phys. Rev. B 95 014306Google Scholar

    [44]

    Kong X, He J, Liang Y, Kou S 2017 Phys. Rev. A 95 33629Google Scholar

    [45]

    Zhu Y Q, Zhang D W, Yan H, Xing D Y, Zhu S L 2017 Phys. Rev. A 96 033634Google Scholar

    [46]

    Tan X, Zhang D W, Liu Q, Xue G, Yu H F, Zhu Y Q, Yan H, Zhu S L, Yu Y 2018 Phys. Rev. Lett. 120 130503Google Scholar

    [47]

    Liang L, Yu Y 2016 Phys. Rev. B 93 045113Google Scholar

    [48]

    Lan Z, Goldman N, Bermudez A, Lu W, Öhberg P 2011 Phys. Rev. B 84 165115Google Scholar

    [49]

    Kitaev A, Laumann C 2009 arXiv: 0904.2771

    [50]

    Trebst S, Troyer M, Wang Z, Ludwig A W W 2008 Prog. Theor. Phys. Supp. 176 384Google Scholar

    [51]

    Nayak C, Simon S H, Stern A, Freedman M, Sarma S D 2008 Rev. Mod. Phys. 80 1083Google Scholar

    [52]

    Read N, Rezayi E 1999 Phys. Rev. B 59 8084Google Scholar

    [53]

    Liu S, Shan C J, Zhang Z M, Xue Z Y 2014 Quantum Inf. Process. 13 1813Google Scholar

    [54]

    Vaishnav J Y, Clark C W 2008 Phys. Rev. Lett. 100 153002Google Scholar

    [55]

    Zhang D W, Xue Z Y, Yan H, Wang Z D, Zhu S L 2012 Phys. Rev. A 85 013628Google Scholar

    [56]

    Li Z, Wang H Q, Zhang D W, Zhu S L, Xing D Y 2016 Phys. Rev. A 94 043617Google Scholar

    [57]

    Xu Y, Duan L M 2017 Phys. Rev. B 96 155301Google Scholar

    [58]

    Shen X, Zhu Y Q, Li Z (In preparation)

    [59]

    Bliokh K Y, Smirnova D, Nori F 2015 Science 348 1448Google Scholar

    [60]

    邱英, 何军, 王彦华, 王婧, 张天才, 王军民 2008 物理学报 57 6227Google Scholar

    Qiu Y, He J, Wang Y H, Wang J, Zhang T C, Wang J M 2008 Acta Phys. Sin. 57 6227Google Scholar

    [61]

    Atala M, Aidelsburger M, Barreiro J T, Abanin D, Kitagawa T, Demler E, Bloch I 2013 Nat. Phys. 9 795Google Scholar

    [62]

    Fisher M P A, Weichwan P B, Grinstein G, Fisher D S 1989 Phys. Rev. B 40 546Google Scholar

    [63]

    Jaksch D, Bruder C, Cirac J I, Zoller P 1998 Phys. Rev. Lett. 81 3108Google Scholar

    [64]

    Jotzu G, Messer M, Desbuquois R, Lebrat M, Uehlinger T, Greif D, Esslinger T 2014 Nature 515 237Google Scholar

    [65]

    Aidelsburger M, Lohse M, Schweizer C, Atala M, Barreiro J T, Nascimbène S, Cooper N R, Bloch I, Goldman N 2015 Nat. Phys. 11 162Google Scholar

    [66]

    杨圆, 陈帅, 李小兵 2018 物理学报 67 237101Google Scholar

    Yang Y, Chen S, Li X B 2018 Acta Phys. Sin. 67 237101Google Scholar

    [67]

    范桁 2018 物理学报 67 120301Google Scholar

    Fan H 2018 Acta Phys. Sin. 67 120301Google Scholar

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Publishing process
  • Received Date:  30 October 2018
  • Accepted Date:  19 January 2019
  • Available Online:  01 February 2019
  • Published Online:  20 February 2019

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