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The quantum walk of two hard-core bosons in one-dimensional lattice under the effect of long-range inter-particle interaction is studied in detail. We also simulate the influence of an isolated defect that may exist in the lattice on the quantum walk of two particles by adding an additional potential energy to a certain lattice site. Using exact diagonalization method, the continuous-time quantum walk is directly simulated. The numerical simulations show that the range of interaction (long-range or short-range), the strength of the inter-particle interaction, the initial state of the two particles and the presence of the isolated defect have great influences on the quantum walk. Under the effect of strong long-range interaction, the particles initially located on the non-adjacent lattice sites have a co-walking behavior, while under the short-range interactions (nearest-neighbor interactions) only two particles initially located on the neighboring lattice sites can exhibit co-walking. After introducing the isolated defect into the system with strong interaction, two particles residing on the same side of the isolated defect keep co-walking, while two particles located on either sides of the isolated defect or one particle located on the isolated defect and the other particle staying on the side of the isolated defect, the two particles keep stationary or co-walking near the defect, displaying the characteristics of localization. By using the second-order perturbation theory of degenerate quantum system, a comprehensive theoretical analysis of the above numerical results is given. The theoretical analysis reveals the underlying physical law of quantum walks of two particles in one-dimensional lattice under the effects of strong long-range interaction and isolated defect in the lattice.
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Keywords:
- quantum walk /
- long-range interactions /
- spatial correlation function
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[11] Gilead Y, Verbin M, Silberberg Y 2015 Phys. Rev. Lett. 115 133602
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[12] Feldker T, Bachor P, Stappel M, Kolbe D, Gerritsma R, Walz J, Schmidt-Kaler F 2015 Phys. Rev. Lett. 115 173001
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[14] Fukuhara T, Kantian A, Endres M, Cheneau M, Schauß P, Hild S, D. Bellem, Schollwöck U, Giamarchi T, Gross C, Bloch I, Kuhr S 2013 Nat. Phys. 9 235
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Google Scholar
[18] Wang L M, Wang L, Zhang Y B 2014 Phys. Rev. A 90 063618
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[20] Sowinski T, Dutta O, Hauke P, Tagliacozzo L, Lewenstein M 2012 Phys. Rev. Lett. 108 115301
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[21] Yan B, Moses S A, Gadway B, Covey J P, Hazzard K R A, Rey A M, Jin D S, Ye J 2013 Nature 501 521
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Google Scholar
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Google Scholar
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图 1 初始处于量子态
${\left| \varPsi \right\rangle _{\text{i}}} = \left| {12, 17} \right\rangle $ 在不同强度的长程相互作用下的两个硬核玻色子的空间关联函数随时间的变化. 图中每一列均表示在特定条件下两粒子关联图样随时间的变化. 从第1行到第3行对应的演化时间$Jt = $ 2, 10, 100. 从第1列到第7列相互作用强度$U/J$ 值分别为$0,\; 2, \;-2,\; 40, \;-40, \;80,\; -80$ . 计算中晶格总的数目为$L = 31$ Figure 1. Spatial pair correlations of two hard-core bosons under the effects of long-range inter-particle interactions in Eq. (3) with various values of the interaction strength. The initial state for the two bosons is
${\left| \varPsi \right\rangle _{\text{i}}} = \left| {12, 17} \right\rangle $ for all the cases. Each column in the figure represents the change of two-particle correlation pattern with time under specific conditions. From top to bottom, the corresponding evolution times are$Jt = $ 2, 10, and 100, respectively. For each column (from left to right) the interaction strength is$U/J = 0, \;2,\; -2, \;40, \;-40, \;80, $ and$-80$ , respectively. The total number of lattice sites is$L = 31$ .
图 2 初始处于不同的量子态的两个硬核玻色子在强长程相互作用下的空间关联图样随时间的变化. 图中每一列表示在特定条件下两粒子关联图样随时间的变化. 从第1行到第4行对应的演化时间分别为
$Jt = 0, {\text{ }}10, {\text{ }}40, {\text{ }}110$ . 从第1列到第6列(从左到右)两粒子的初始态${\left| \varPsi \right\rangle _{\text{i}}}$ 分别为$\left| {15, 17} \right\rangle $ ,$\left| {14, 18} \right\rangle $ ,$\left| {13, 19} \right\rangle $ ,$\left| {8, 24} \right\rangle $ ,$\left| {4, 28} \right\rangle $ ,$\left| {8, 24} \right\rangle $ . 虚线左侧$U/J = 80$ , 虚线右侧$U/J = 1{\text{ }}000$ . 计算中晶格格点总的数目为$L = 31$ Figure 2. Spatial pair correlations of two hard-core bosons under the effects of strong long-range inter-particle interactions for different initial states. Each column in the figure represents the change of two-particle correlation pattern with time under specific conditions. From top to bottom, the corresponding evolution times are
$Jt = 0, {\text{ }}10, {\text{ }}40$ , and$110$ , respectively. The initial states of the two particles${\left| \varPsi \right\rangle _{\text{i}}}$ in each column (from left to right) are$\left| {15, 17} \right\rangle $ ,$\left| {14, 18} \right\rangle $ ,$\left| {13, 19} \right\rangle $ ,$\left| {8, 24} \right\rangle $ ,$\left| {4, 28} \right\rangle $ , and$\left| {8, 24} \right\rangle $ , respectively. On the left and right size of the dashed line the interaction strength is$U/J = 80$ and$1{\text{ }}000$ , respectively. The total number of lattice sites is$L = 31$ .
图 3 初始处于不同量子态的两个硬核玻色子在强短程相互作用下(虚线左侧)和无相互作用下(虚线右侧)的空间关联函数图样随时间的变化. 对于强短程相互作用情形, 相互作用强度为
$U/J = 80$ . 从第1排到第4排对应的演化时间分别为$Jt = 0, $ $ 10, {\text{ }}80, {\text{ }}150$ . 从第1列到第4列(从左到右)两粒子的初始态${\left| \varPsi \right\rangle _{\text{i}}}$ 为$\left| {15, 17} \right\rangle $ ,$\left| {10, 22} \right\rangle $ ,$\left| {15, 17} \right\rangle $ ,$\left| {10, 22} \right\rangle $ . 计算中晶格格点总的数目为$L = 31$ Figure 3. Spatial pair correlations of two hard-core bosons under the effects of strong short-range inter-particle interactions in Eq. (2) (left side of the dotted line) and no interactions (right side of the dotted line) for different initial states. The interaction strength for the short-range inter-particle interactions is
$U/J = 80$ . From top to bottom, the corresponding evolution times are$Jt = 0, {\text{ }}10, {\text{ }}80, {\text{ }}$ and$150$ , respectively. The initial state of the two particles${\left| \varPsi \right\rangle _{\text{i}}}$ in each column (from left to right) is$\left| {15, 17} \right\rangle $ ,$\left| {10, 22} \right\rangle $ ,$\left| {15, 17} \right\rangle $ , and$\left| {10, 22} \right\rangle $ , respectively. The total number of lattice sites is$L = 31$ .
图 4 在具有孤立缺陷点的晶格势阱中, 处于不同初始态的两个硬核玻色子在强长程(虚线左侧)和短程相互作用(虚线右侧)下空间关联函数随时间的变化. 晶格中孤立缺陷点按如下方式引入: 处于在总数为
$L = 31$ 个格点的晶格中设定处于格点17的粒子具有额外的势能, 设哈密顿量(1)式中$V\left( {17} \right)/J = - 80$ , 而其他格点的局域势能仍为0.长程相互作用与短程相互作用的作用强度都为$U/J = 80$ . 第1行到第4行(从上到下)对应的演化时间分别为$Jt = 0, {\text{ }}10, {\text{ }}80, {\text{ }}200$ . 从第1列到第6列对应的初始量子态${\left| \varPsi \right\rangle _{\text{i}}}$ 为$\left| {10, 12} \right\rangle $ ,$\left| {16, 18} \right\rangle $ ,$\left| {15, 18} \right\rangle $ ,$\left| {15, 17} \right\rangle $ ,$\left| {16, 18} \right\rangle $ ,$\left| {16, 17} \right\rangle $ Figure 4. Spatial pair correlations of two hard-core bosons under the effects of an isolated defect point in the lattice and strong inter-particle interactions for different initial states. The left and right size of the dashed line are under the effects of strong long-range and short-range inter-particle interactions, respectively. The isolated defect point is introduced as follows: in the lattice with
$L = 31$ lattice sites in total, lattice site 17 is set to have additional potential energy with strength$V\left( {17} \right)/J = - 80$ in the Hamiltonian (1), while the local potential energy of the other lattice sites is still 0. The interaction strength is$U/J = 80$ for all cases. From top to bottom, the corresponding evolution times are$Jt = 0, {\text{ }}10, {\text{ }}80$ and$200$ , respectively. The initial state of two particles${\left| \varPsi \right\rangle _{\text{i}}}$ in each column (from left to right) is$\left| {10, 12} \right\rangle $ ,$\left| {16, 18} \right\rangle $ ,$\left| {15, 18} \right\rangle $ ,$\left| {15, 17} \right\rangle $ ,$\left| {16, 18} \right\rangle $ , and$\left| {16, 17} \right\rangle $ , respectively. -
[1] Lovasz L 1989 Bol. Soc. Bras. Mat. 20 87
Google Scholar
[2] Adamic L A, Lukose R M, Puniyani A R, Huberman B A 2001 Phys. Rev. E 64 046135
Google Scholar
[3] Noh J D, Rieger H 2004 Phys. Rev. Lett. 92 118701
Google Scholar
[4] Yin R, Barkai E 2023 Phys. Rev. Lett. 130 050802
Google Scholar
[5] Lewis D, Benhemou A, Feinstein N, Banchi L, Bose S 2021 Phys. Rev. Lett. 126 240502
Google Scholar
[6] Lovett N B, Cooper S, Everitt M, Trevers M, Kendon V 2010 Phys. Rev. A 81 042330
Google Scholar
[7] Childs A M, Gosset D, Webb Z 2013 Science 339 791
Google Scholar
[8] Farhi E, Gutmann S 1998 Phys. Rev. A 58 915
Google Scholar
[9] Childs A M, Cleve R, Deotto E, Farhi E, Gutmann S, Spielman D A 2003 Proceedings of the thirty-fifth ACM Symposium on Theory of Computing San Diego, CA, USA, June 9–11, 2003 (New York: ACM Press
[10] Crespi A, Osellame R, Ramponi R, Giovannetti V, Fazio R, Sansoni L, Nicola F De, Sciarrino F, Mataloni P 2013 Nat. Photonics 7 322
Google Scholar
[11] Gilead Y, Verbin M, Silberberg Y 2015 Phys. Rev. Lett. 115 133602
Google Scholar
[12] Feldker T, Bachor P, Stappel M, Kolbe D, Gerritsma R, Walz J, Schmidt-Kaler F 2015 Phys. Rev. Lett. 115 173001
Google Scholar
[13] Preiss P M, Ma R, Tai M E, Lukin A, Rispoli M, Zupancic P, Lahini Y, Islam R, Greiner M 2015 Science 347 1229
Google Scholar
[14] Fukuhara T, Kantian A, Endres M, Cheneau M, Schauß P, Hild S, D. Bellem, Schollwöck U, Giamarchi T, Gross C, Bloch I, Kuhr S 2013 Nat. Phys. 9 235
Google Scholar
[15] Du J, Li H, Xu X, Shi M, Wu J, Zhou X, Han R 2003 Phys. Rev. A 67 042316
Google Scholar
[16] Kempe J 2003 Contemp. Phys. 44 307
Google Scholar
[17] Qin X Z, Ke Y G, Guan X W, Li Z B, Andrei N, Lee C H 2014 Phys. Rev. A 90 062301
Google Scholar
[18] Wang L M, Wang L, Zhang Y B 2014 Phys. Rev. A 90 063618
Google Scholar
[19] Jurcevic P, Lanyon B P, Hauke P, Hempel C, Zoller P, Blatt R, Roos C F 2014 Nature 511 202
Google Scholar
[20] Sowinski T, Dutta O, Hauke P, Tagliacozzo L, Lewenstein M 2012 Phys. Rev. Lett. 108 115301
Google Scholar
[21] Yan B, Moses S A, Gadway B, Covey J P, Hazzard K R A, Rey A M, Jin D S, Ye J 2013 Nature 501 521
Google Scholar
[22] Lang F, Winkler K, Strauss C, Grimm R, Hecker Denschlag J 2008 Phys. Rev. Lett. 101 133005
Google Scholar
[23] Carr L D, DeMille D, Krems R V, Ye J 2009 New J. Phys. 11 055049
Google Scholar
[24] Ni K K, Rosenband T, Grimes D D 2018 Chem. Sci. 9 6830
Google Scholar
[25] Kaufman A M, Ni K K 2021 Nat. Phys. 17 1324
Google Scholar
[26] Ho C, Devlin J, Rabey I, Yzombard P, Lim J, Wright S, Fitch N, Hinds E, Tarbutt M, Sauer B 2020 New J. Phys. 22 053031
Google Scholar
[27] Liu Y, Hu M G, Nichols M A, Yang D, Xie D, Guo H, Ni K K 2021 Nature 593 379
Google Scholar
[28] Liu Y and Ni K K 2022 Annu. Rev. Phys. Chem. 73 73
Google Scholar
[29] S. Fölling, Gerbier F, Widera A, Mandel O, Gericke T, Bloch I 2005 Nature 434 481
Google Scholar
[30] Greiner M, Regal C A, Stewart J T, Jin D S 2005 Phys. Rev. Lett. 94 110401
Google Scholar
[31] Schiulaz M, Silva A, Müller M 2015 Phys. Rev. B 91 184202
Google Scholar
[32] Michal V P, Altshuler B L, Shlyapnikov G V 2014 Phys. Rev. Lett. 113 045304
Google Scholar
[33] Anderson P W 1958 Phys. Rev. 109 1492
Google Scholar
[34] Vu D D, Huang K, Li X, Das Sarma S 2022 Phys. Rev. Lett. 128 146601
Google Scholar
[35] Xue P, Zhang R, Bian Z H, Zhan X, Qin H, Sanders B C 2015 Phys. Rev. A 92 042316
Google Scholar
[36] Takahashi M 1977 J. Phys. C Solid State Phys. 10 1289
Google Scholar
[37] Liu W, Andrei N 2014 Phys. Rev. Lett. 112 257204
Google Scholar
[38] Chattaraj T, Krems R V 2016 Phys. Rev. A 94 023601
Google Scholar
[39] Nandkishore R, Huse D A 2015 Annu. Rev. Condens. Matter Phys. 6 15
Google Scholar
[40] Abanin D A, Altman E, Bloch I, Serbyn M 2019 Rev. Modern Phys. 91 021001
Google Scholar
[41] Schulz M, Hooley C A, Moessner R, Pollmann F 2019 Phys. Rev. Lett. 122, 040606
Google Scholar
[42] Kuno Y, Orito T, Ichinose I 2020 New J. Phys. 22 013032
Google Scholar
[43] Zhang N, Ke Y, Lin Ling, Zhang L, Lee C H 2023 New J. Phys. 25 043021
Google Scholar
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