Relativistic quantum effects of Dirac particles simulated by ultracold atoms

Abstract

Quantum simulation is a powerful tool to study a variety of problems in physics, ranging from high-energy physics to condensed-matter physics. In this article, we review the recent theoretical and experimental progress in quantum simulation of Dirac equation with tunable parameters by using ultracold neutral atoms trapped in optical lattices or subject to light-induced synthetic gauge fields. The effective theories for the quasiparticles become relativistic under certain conditions in these systems, making them ideal platforms for studying the exotic relativistic effects. We focus on the realization of one, two, and three dimensional Dirac equations as well as the detection of some relativistic effects, including particularly the well-known Zitterbewegung effect and Klein tunneling. The realization of quantum anomalous Hall effects is also briefly discussed.

Keywords

ultracold atoms / Dirac equation / quantum simulation

References

[1] R.Feynman, Int. J. Theor. Phys., 1982, 21(6-7): 467

[2] See the reviews in Insight: Quantum Coherence, Nature, 2008, 453(7198): 1003

[3] I.Buluta and F.Nori, Science, 2009, 326(5949): 108

[4] M.Greiner, O.Mandel, T.Esslinger, T. W.Hänsch, and I.Bloch, Nature, 2002, 415(6867): 39

[5] T.Stöferle, H.Moritz, C.Schori, M.Köhl, and T.Esslinger, Phys. Rev. Lett., 2004, 92(13): 130403

[6] I. B.Spielman, W. D.Phillips, and J. V.Porto, Phys. Rev. Lett., 2007, 98(8): 080404

[7] J.Simon, W. S.Bakr, R.Ma, M. E.Tai, P. M.Preiss, and M.Greiner, Nature, 2011, 472(7343): 307

[8] A.Friedenauer, H.Schmitz, J. T.Glueckert, D.Porras, and T.Schaetz, Nat. Phys., 2008, 4(10): 757

[9] K.Kim, M. S.Chang, S.Korenblit, R.Islam, E. E.Edwards, J. K.Freericks, G. D.Lin, L. M.Duan, and C.Monroe, Nature, 2010, 465(7298): 590

[10] R.Gerritsma, G.Kirchmair, F.Zähringer, E.Solano, R.Blatt, and C. F.Roos, Nature, 2010, 463(7277): 68

[11] R.Gerritsma, B. P.Lanyon, G.Kirchmair, F.Zähringer, C.Hempel, J.Casanova, J. J.Garcá-Ripoll, E.Solano, R.Blatt, and C. F.Roos, Phys. Rev. Lett., 2011, 106(6): 060503

[12] J. T.Barreiro, M.M¨uler, P.Schindler, D.Nigg, T.Monz, M.Chwalla, M.Hennrich, C. F.Roos, P.Zoller, and R.Blatt, Nature, 2011, 470(7335): 486

[13] J.Du, N.Xu, X.Peng, P.Wang, S.Wu, and D.Lu, Phys. Rev. Lett., 2010, 104(3): 030502

[14] M. H.Anderson, J. R.Ensher, M. R.Matthews, C. E.Wieman, and E. A.Cornell, Science, 1995, 269(5221): 198

[15] F.Dalfovo, S.Giorgini, L. P.Pitaevskii, and S.Stringari, Rev. Mod. Phys., 1999, 71(3): 463

[16] C. A.Regal, M.Greiner, and D. S.Jin, Phys. Rev. Lett., 2004, 92(4): 040403

[17] S.Giorgini, L. P.Pitaevskii, and S.Stringari, Rev. Mod. Phys., 2008, 80(4): 1215

[18] J.Liu and B.Liu, Front. Phys. China, 2010, 5(2): 123

[19] H.Jing, Y.Jiang, and Y.Deng, Front. Phys., 2011, 6(1): 15

[20] M.Lewenstein, A.Sanpera, V.Ahufinger, B.Damski, A.Sende, and U.Sen, Adv. Phys., 2007, 56(2): 243

[21] H.Zhai, Front. Phys. China, 2009, 4(1): 1

[22] A.Klein and D.Jaksch, Phys. Rev. A, 2006, 73(5): 053613

[23] M. A.Baranov, K.Osterloh, and M.Lewenstein, Phys. Rev. Lett., 2005, 94(7): 070404

[24] A. S.Sørensen, E.Demler, and M. D.Lukin, Phys. Rev. Lett., 2005, 94(8): 086803

[25] G. B.Jo, Y. R.Lee, J. H.Choi, C. A.Christensen, T. H.Kim, J. H.Thywissen, D. E.Pritchard, and W.Ketterle, Science, 2009, 325(5947): 1521

[26] J.Billy, V.Josse, Z.Zuo, A.Bernard, B.Hambrecht, P.Lugan, D.Cléent, L.Sanchez-Palencia, P.Bouyer, and A.Aspect, Nature, 2008, 453(7197): 891

[27] G.Roati, C.D’Errico, L.Fallani, M.Fattori, C.Fort, M.Zaccanti, G.Modugno, M.Modugno, and M.Inguscio, Nature, 2008, 453(7197): 895

[28] B.Deissler, M.Zaccanti, G.Roati, C.D’Errico, M.Fattori, M.Modugno, G.Modugno, and M.Inguscio, Nat. Phys., 2010, 6(5): 354

[29] L.Sanchez-Palencia and M.Lewenstein, Nat. Phys., 2010, 6(2): 87

[30] L. J.Garay, J. R.Anglin, J. I.Cirac, and P.Zoller, Phys. Rev. Lett., 2000, 85(22): 4643

[31] U. R.Fischer and R.Schützhold, Phys. Rev. A, 2004, 70(6): 063615

[32] M.Snoek, M.Haque, S.Vandoren, and H. T. C.Stoof, Phys. Rev. Lett., 2005, 95(25): 250401

[33] Y.Yu and K.Yang, Phys. Rev. Lett., 2008, 100(9): 090404

[34] Y.Yu and K.Yang, Phys. Rev. Lett., 2010, 105(15): 150605

[35] K. S.Novoselov, A. K.Geim, S. V.Morozov, D.Jiang, Y.Zhang, S. V.Dubonos, I. V.Grigorieva, and A. A.Firsov, Science, 2004, 306(5696): 666

[36] K. S.Novoselov, A. K.Geim, S. V.Morozov, D.Jiang, M. I.Katsnelson, I. V.Grigorieva, S. V.Dubonos, and A. A.Firsov, Nature, 2005, 438(7065): 197

[37] M. Z.Hasan and C. L.Kane, Rev. Mod. Phys., 2010, 82(4): 3045

[38] W.Greiner, Relativistic Quantum Mechanics, 3rd Ed., Berlin: Spinger-Verleg, 2003

[39] E.Schrödinger, Sitzungsber. Preuss. Akad. Wiss. Phys.-Math. Kl., 1930, 24: 418

[40] O.Klein, Z. Phys., 1929, 50: 157

[41] S. L.Zhu, B.Wang, and L. M.Duan, Phys. Rev. Lett., 2007, 98(26): 260402

[42] E.Zhao and A.Paramekanti, Phys. Rev. Lett., 2006, 97(23): 230404

[43] C.Wu and S.Das Sarma, Phys. Rev. B, 2008, 77(23): 235107

[44] B.Wunsch, F.Guinea, and F.Sols, New J. Phys., 2008, 10(10): 103027

[45] K. L.Lee, B.Grémaud, R.Han, B. G.Englert, and C.Miniatura, Phys. Rev. A, 2009, 80(4): 043411

[46] A.Dutta, R. R. P.Singh, and U.Divakaran, Europhys. Lett., 2010, 89(6): 67001

[47] D.Poletti, C.Miniatura, and B.Grémaud, Europhys. Lett., 2011, 93(3): 37008

[48] D.Bercioux, D. F.Urban, H.Grabert, and W.Häusler, Phys. Rev. A, 2009, 80(6): 063603

[49] D.Bercioux, N.Goldman, and D. F.Urban, Phys. Rev. A, 2011, 83(2): 023609

[50] R.Shen, L. B.Shao, B.Wang, and D. Y.Xing, Phys. Rev. B, 2010, 81(4): 041410 (R)

[51] I. I.Satija, D. C.Dakin, J. Y.Vaishnav, and C. W.Clark, Phys. Rev. A, 2008, 77(4): 043410

[52] J. M.Hou, W. X.Yang, and X. J.Liu, Phys. Rev. A, 2009, 79(4): 043621

[53] L. K.Lim, C. M.Smith, and A.Hemmerich, Phys. Rev. Lett., 2008, 100(13): 130402

[54] L. K.Lim, A.Hemmerich, and C. M.Smith, Phys. Rev. A, 2010, 81(2): 023404

[55] L. K.Lim, A.Lazarides, A.Hemmerich, and C. M.Smith, Europhys. Lett., 2009, 88(3): 36001

[56] N.Goldman, A.Kubasiak, A.Bermudez, P.Gaspard, M.Lewenstein, and M. A.Martin-Delgado, Phys. Rev. Lett., 2009, 103(3): 035301

[57] X. J.Liu, X.Liu, C.Wu, and J.Sinova, Phys. Rev. A, 2010, 81(3): 033622

[58] M. P.Kennett, N.Komeilizadeh, K.Kaveh, and P. M.Smith, Phys. Rev. A, 2011, 83(5): 053636

[59] M.Yang and S. L.Zhu, Phys. Rev. A, 2010, 82(6): 064102

[60] A.Bermudez, L.Mazza, M.Rizzi, N.Goldman, M.Lewenstein, and M. A.Martin-Delgado, Phys. Rev. Lett., 2010, 105(19): 190404

[61] L.Lepori, G.Mussardo, and A.Trombettoni, Europhys. Lett., 2010, 92(5): 50003

[62] L.M.Duan, E.Demler, and M. D.Lukin, Phys. Rev. Lett., 2003, 91(9): 090402

[63] Y.Zhang, Y. W.Tan, H. L.Stormer, and P.Kim, Nature, 2005, 438(7065): 201

[64] G. W.Semenoff, Phys. Rev. Lett., 1984, 53(26): 2449

[65] X. G.Wen, Quantum Field Theory of Many-Body Systems, Oxford: Oxford University, 2004

[66] A.Bermudez, N.Goldman, A.Kubasiak, M.Lewenstein, and M. A.Martin-Delgado, New J. Phys., 2010, 12(3): 033041

[67] J. K.Block and N.Nygaard, Phys. Rev. A, 2010, 81(5): 053421

[68] C.Wu, D.Bergman, L.Balents, and S.Das Sarma, Phys. Rev. Lett., 2007, 99(7): 070401

[69] P.Soltan-Panahi, J.Struck, P.Hauke, A.Bick, W.Plenkers, G.Meineke, C.Becker, P.Windpassinger, M.Lewenstein, and K.Sengstock, Nat. Phys., 2011, 7(5): 434

[70] V.Apaja, M.Hyrkäs, and M.Manninen, Phys. Rev. A, 2010, 82(4): 041402 (R)

[71] D.Jaksch and P.Zoller, New J. Phys., 2003, 5(1): 56

[72] K.Osterloh, M.Baig, L.Santos, P.Zoller, and M.Lewenstein, Phys. Rev. Lett., 2005, 95(1): 010403

[73] F.Gerbier and J.Dalibard, New J. Phys., 2010, 12(3): 033007

[74] N.Goldman, A.Kubasiak, P.Gaspard, and M.Lewenstein, Phys. Rev. A, 2009, 79(2): 023624

[75] L. H.Karsten and J.Smith, Nucl. Phys. B, 1981, 183(1-2): 103

[76] H. B.Nielsen and M.Ninomiya, Nucl. Phys. B, 1981, 185(1): 20

[77] K.Wilson, New Phenomena in Subnuclear Physics, New York: Plenum, 1977

[78] X. L.Qi, R.Li, J.Zang, and S. C.Zhang, Science, 2009, 323(5918): 1184

[79] M. W.Zwierlein, A.Schirotzek, C. H.Schunck, and W.Ketterle, Science, 2006, 311(5760): 492

[80] J.Stenger, S.Inouye, A. P.Chikkatur, D. M.Stamper-Kurn, D. E.Pritchard, and W.Ketterle, Phys. Rev. Lett., 1999, 82(23): 4569

[81] F. D. M.Haldane, Phys. Rev. Lett., 1988, 61(18): 2015

[82] M.Onoda and N.Nagaosa, Phys. Rev. Lett., 2003, 90(20): 206601

[83] C. X.Liu, X. L.Qi, X.Dai, Z.Fang, and S. C.Zhang, Phys. Rev. Lett., 2008, 101(14): 146802

[84] R.Yu, W.Zhang, H. J.Zhang, S. C.Zhang, X.Dai, and Z.Fang, Science, 2010, 329(5987): 61

[85] Z.Qiao, S. A.Yang, W.Feng, W. K.Tse, J.Ding, Y.Yao, J.Wang, and Q.Niu, Phys. Rev. B, 2010, 82(16): 161414

[86] Y.Zhang and C.Zhang, Phys. Rev. B, 2011, 84(8): 085123

[87] H.Li, L.Sheng, and D. Y.Xing, Phys. Rev. B, 2011, 84(3): 035310

[88] L. B.Shao, S. L.Zhu, L.Sheng, D. Y.Xing, and Z. D.Wang, Phys. Rev. Lett., 2008, 101(24): 246810

[89] C.Wu, Phys. Rev. Lett., 2008, 101(18): 186807

[90] N.Gemelke, <DissertationTip/>, Stanford University, 2007

[91] M.Zhang, H.Hung, C.Zhang, and C.Wu, Phys. Rev. A, 2011, 83(2): 023615

[92] X. J.Liu, X.Liu, C.Wu, and J.Sinova, Phys. Rev. A, 2010, 81(3): 033622

[93] G.Juzeliunas and P.Öhberg, Phys. Rev. Lett., 2004, 93(3): 033602

[94] G.Juzeliunas, P.Öhberg, J.Ruseckas, and A.Klein, Phys. Rev. A, 2005, 71(5): 053614

[95] P.Öhberg, G.Juzeliunas, J.Ruseckas, and M.Fleischhauer, Phys. Rev. A, 2005, 72(5): 053632

[96] G.Juzeliunas, J.Ruseckas, P.Öhberg, and M.Fleischhauer, Phys. Rev. A, 2006, 73(2):025602

[97] S. L.Zhu, H.Fu, C. J.Wu, S. C.Zhang, and L. M.Duan, Phys. Rev. Lett., 2006, 97(24): 240401

[98] K. J.Güunter, M.Cheneau, T.Yefsah, S. P.Rath, and J.Dalibard, Phys. Rev. A, 2009, 79(1): 011604(R)

[99] J.Dalibard, F.Gerbier, G.Juzeliunas, and P.Öhberg, arXiv:cond-mat/1008.5378, 2010

[100] J.Ruseckas and G.Juzeliunas, Phys. Rev. Lett., 2005, 95(1): 010404

[101] F.Wilczek and A.Zee, Phys. Rev. Lett., 1984, 52(24): 2111

[102] G.Juzeliunas, J.Ruseckas, M.Lindberg, L.Santos, and P.Öhberg, Phys. Rev. A, 2008, 77(1): 011802(R)

[103] S. L.Zhu, D. W.Zhang, and Z. D.Wang, Phys. Rev. Lett., 2009, 102(21): 210403

[104] M.Merkl, F. E.Zimmer, G.Juzeliunas, and P.Öhberg, Europhys. Lett., 2008, 83(5): 54002

[105] D. W.Zhang, Z. Y.Xue, H.Yan, Z. D.Wang, and S. L.Zhu, arXiv:1104.0444, 2011

[106] N. R.Cooper, Adv. Phys., 2008, 57(6): 539

[107] M.Burrello and A.Trombettoni, Phys. Rev. Lett., 2010, 105(12): 125304

[108] X. J.Liu, X.Liu, L. C.Kwek, and C. H.Oh, Front. Phys. China, 2008, 3(2): 113

[109] I. B.Spielman, Phys. Rev. A, 2009, 79(6): 063613

[110] Y. J.Lin, R. L.Compton, A. R.Perry, W. D.Phillips, J. V.Porto, and I. B.Spielman, Phys. Rev. Lett., 2009, 102(13): 130401

[111] Y. J.Lin, R. L.Compton, K.Jim´enez-Garćıa, J. V.Porto, and I. B.Spielman, Nature, 2009, 462(7273): 628

[112] Y. J.Lin, R. L.Compton, K.Jim´enez-Garćıa, W. D.Phillips, J. V.Porto, and I. B.Spielman, Nat. Phys., 2011, 7: 531

[113] T. D.Stanescu, B.Anderson, and V.Galitski, Phys. Rev. A, 2008, 78(2): 023616

[114] C. J.Wang, C.Gao, C. M.Jian, and H.Zhai, Phys. Rev. Lett., 2010, 105(16): 160403

[115] T. L.Ho and S.Zhang, arXiv:cond-mat/1007.0650, 2010

[116] Y. J.Lin, K.Jim´enez-Garćıa, and I. B.Spielman, Nature, 2011, 471(7336): 83

[117] X. J.Liu, M. F.Borunda, X.Liu, and J.Sinova, Phys. Rev. Lett., 2009, 102(4): 046402

[118] G.Juzeliunas, J.Ruseckas, and J.Dalibard, Phys. Rev. A, 2010, 81(5): 053403

[119] Z.Lan, N.Goldman, A.Bermudez, W.Lu, and P.Öhberg, Phys. Rev. B, 2011, 84(16): 165115

[120] M.Merkl, A.Jacob, F. E.Zimmer, P.Öhberg, and L.Santos, Phys. Rev. Lett., 2010, 104(7): 073603

[121] L. H.Haddad and L. D.Carr, Physica D, 2009, 238(15): 1413

[122] L. H.Haddad and L. D.Carr, Europhys. Lett., 2011, 94(5): 56002

[123] Z.Chen and B.Wu, Phys. Rev. Lett., 2011, 107(6): 065301

[124] See the review, W.Zawadzki and T. M.Rusin, J. Phys.: Condens. Matter, 2011, 23(14): 143201

[125] J.Schliemann, D.Loss, and R. M.Westervelt, Phys. Rev. Lett., 2005, 94(20): 206801

[126] M. I.Katsnelson, Eur. Phys. J. B, 2006, 51(2): 157

[127] B.Trauzettel, Y. M.Blanter, and A. F.Morpurgo, Phys. Rev. B, 2007, 75(3): 035305

[128] D.Lurié and S.Cremer, Physica, 1970, 50(2): 224

[129] X.Zhang, Phys. Rev. Lett., 2008, 100(11): 113903

[130] F.Dreisow, M.Heinrich, R.Keil, A.T¨unnermann, S.Nolte, S.Longhi, and A.Szameit, Phys. Rev. Lett., 2010, 105(14): 143902

[131] L.Lamata, J.León, T.Sch¨atz, and E.Solano, Phys. Rev. Lett., 2007, 98(25): 253005

[132] J. Y.Vaishnav and C. W.Clark, Phys. Rev. Lett., 2008, 100(15):

[133] J. J.Song and B. A.Foreman, Phys. Rev. A, 2009, 80(4): 045602

[134] Q.Zhang, J.Gong, and C. H.Oh, Phys. Rev. A, 2010, 81(2): 023608

[135] D.Witthaut, Phys. Rev. A, 2010, 82(3): 033602

[136] J.Larson, J. P.Martikainen, A.Collin, and E.Sjöqvist, Phys. Rev. A, 2010, 82(4): 043620

[137] M. I.Katsnelson, K. S.Novoselov, and A. K.Geim, Nat. Phys., 2006, 2(9): 620

[138] O.Bahat-Treidel, O.Peleg, M.Grobman, N.Shapira, M.Segev, and T.Pereg-Barnea, Phys. Rev. Lett., 2010, 104(6): 063901

[139] D.Witthaut, T.Salger, S.Kling, C.Grossert, and M.Weitz, Phys. Rev. A, 2011, 84(3): 033601

[140] T.Salger, C.Grossert, S.Kling, and M.Weitz, arXiv:condmat/1108.4447, 2011

[141] J.Otterbach, R. G.Unanyan, and M.Fleischhauer, Phys. Rev. Lett., 2009, 102(6): 063602

[142] M. G.Tarallo, J.Miller, J.Agresti, E.D’Ambrosio, R.De-Salvo, D.Forest, B.Lagrange, J. M.Mackowsky, C.Michel, J. L.Montorio, N.Morgado, L.Pinard, A.Remilleux, B.Simoni, and P.Willems, Appl. Opt., 2007, 46(26): 6648

[143] L.Khaykovich, F.Schreck, G.Ferrari, T.Bourdel, J.Cubizolles, L. D.Carr, Y.Castin, and C.Salomon, Science, 2002, 296(5571): 1290

[144] N.Dombey, P.Kennedy, and A.Calogeracos, Phys. Rev. Lett., 2000, 85(9): 1787

[145] F.Wilczek, Nat. Phys., 2009, 5(9): 614

[146] R. F.Service, Science, 2011, 332(6026): 193

[147] S. L.Zhu, L. B.Shao, Z. D.Wang, and L. M.Duan, Phys. Rev. Lett., 2011, 106(10): 100404

[148] L.Jiang, T.Kitagawa, J.Alicea, A. R.Akhmerov, D.Pekker, G.Refael, J. I.Cirac, E.Demler, M. D.Lukin, and P.Zoller, Phys. Rev. Lett., 2011, 106(22): 220402

[149] S.Tewari and S.Das Sarma, Phys. Rev. Lett., 2007, 98(1): 010506

[150] C.Nayak, S. H.Simon, A.Stern, M.Freedman, and S.Das Sarma, Rev. Mod. Phys., 2008, 80(3): 1083

[151] J.Alicea, Y.Oreg, G.Refael, F.von Oppen, and M. P. A.Fisher, Nat. Phys., 2011, 7(5): 412

[152] J.Casanova, C.Sabin, J.Leon, I. L.Egusquiza, R.Gerritsma, C. F.Roos, J. J.Garcia-Ripoll, and E.Solano, arXiv:1102.1651, 2011

[153] J. I.Cirac, P.Maraner, and J. K.Pachos, Phys. Rev. Lett., 2010, 105(19): 190403

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