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14 - Preparing and Probing Chern Bands with Cold Atoms

from Part III - Condensates in Atomic Physics

Published online by Cambridge University Press:  18 May 2017

By
N. Goldman ,
N. R. Cooper  and
J. Dalibard
Edited by
Nick P. Proukakis ,
David W. Snoke  and
Peter B. Littlewood
N. Goldman
Affiliation:
Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles
N. R. Cooper
Affiliation:
T.C.M. Group, Cavendish Laboratory, J.J. Thomson Avenue, Cambridge CB3 0HE, UK
J. Dalibard
Affiliation:
Laboratoire Kastler Brossel, Coll`ege de France, CNRS, ENS-PSL Research University
Nick P. Proukakis
Affiliation:
Newcastle University
David W. Snoke
Affiliation:
University of Pittsburgh
Peter B. Littlewood
Affiliation:
University of Chicago
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Summary

The present chapter discusses methods by which topological Bloch bands can be prepared in cold-atom setups. Focusing on the case of Chern bands for two-dimensional systems, we describe how topological properties can be triggered by driving atomic gases, either by dressing internal levels with light or through time-periodic modulations. We illustrate these methods with concrete examples, and we discuss recent experiments where geometrical and topological band properties have been identified.

Introduction

Ultracold atoms constitute a promising physical platform for the preparation and exploration of novel states of matter [1, 2, 3]. In particular, the engineering of topological band structures with cold-atom systems, together with the capability of tuning interactions between the particles, opens an interesting route toward the realization of intriguing strongly correlated states with topological features, such as fractional topological insulators and quantum Hall liquids [4].

This chapter is dedicated to the preparation and the detection of topological band structures characterized by nonzero Chern numbers [5]. Such Chern bands, which constitute the building blocks for realizing (fractional) Chern insulators [6, 7], arise in two-dimensional (2D) systems presenting time-reversal-symmetry (TRS) breaking effects. For instance, nontrivial Chern bands naturally appear in the Harper-Hofstadter model [8], a lattice penetrated by a uniform flux, where they generalize the (nondispersive) Landau levels to the lattice framework. Additionally, Chern bands also appear in staggered flux configurations, such as in Haldane's honeycomb-lattice model [9] or in lattice systems combining Rashba spin-orbit coupling and Zeeman (exchange) fields.

The atoms being charge neutral, “magnetic” fluxes cannot be simply produced by subjecting optical lattices to “real” magnetic fields. It is the aim of this chapter to review several schemes that have been recently implemented in laboratories with the goal of realizing synthetic magnetic fields leading to Chern bands for cold atoms. Our presentation is complementary to that of Chapter 15, as we focus on synthetic magnetic fields in lattice-based systems for which the flux density can be made very large (with magnetic length comparable to interparticle spacing). The chapter is structured as follows: Section 14.2 describes how the Chern number is related to physical observables defined in a lattice framework. In particular, it clarifies the link between recent Chern-number measurements performed in cold bosonic gases and the more conventional (electronic) quantum Hall effect.

Type
Chapter
Information
Universal Themes of Bose-Einstein Condensation , pp. 274 - 298
Publisher: Cambridge University Press
Print publication year: 2017

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References

[1] Bloch, I., Dalibard, J., and Zwerger, W. 2008. Many-body physics with ultracold gases. Rev. Mod. Phys., 80, 885.Google Scholar
[2] Dalibard, J., Gerbier, F., Juzeliūnas, G., and Öhberg, P. 2011. Colloquium: artificial gauge potentials for neutral atoms. Rev. Mod. Phys., 83, 1523.Google Scholar
[3] Goldman, N., Juzeliūnas, G., Öhberg, P., and Spielman, I. B. 2014. Light-induced gauge fields for ultracold atoms. Rep. Prog. Phys., 77, 126401.Google Scholar
[4] Cooper, N. R. 2008. Rapidly rotating atomic gases. Advances in Physics, 57, 539.Google Scholar
[5] Bernevig, B. A., and Hughes, T. L. 2013. Topological Insulators and Topological Superconductors. Princeton, NJ: Princeton University Press.
[6] Parameswaran, S. A., Roy, R., and Sondhi, S. L. 2013. Fractional quantum Hall physics in topological flat bands. Comptes Rendus Physique, 14, 816.Google Scholar
[7] Bergholz, E. J., and Liu, Z. 2013. Topological flat band models and fractional Chern insulators. Int. J. Mod. Phys. B, 27, 1330017.Google Scholar
[8] Hofstadter, D. R. 1976. Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields. Phys. Rev. B, 14, 2239.Google Scholar
[9] Haldane, F. D. M. 1988. Model for a quantum Hall effect without Landau levels: condensed-matter realization of the “parity anomaly.” Phys. Rev. Lett., 61, 2015.Google Scholar
[10] Thouless, D. J., Kohmoto, M., Nightingale, M. P., and den Nijs, M. 1982. Quantized Hall conductance in a two-dimensional periodic potential. Phys. Rev. Lett., 49, 405.Google Scholar
[11] Kohmoto, M. 1985. Topological invariant and the quantization of the Hall conductance. Annals of Physics, 160, 343.Google Scholar
[12] Xiao, D., Chang, M.-C., and Niu, Q. 2010. Berry phase effects on electronic properties. Rev. Mod. Phys., 82, 1959.Google Scholar
[13] Price, H. M., and Cooper, N. R. 2012. Mapping the Berry curvature from semiclassical dynamics in optical lattices. Phys. Rev. A, 85, 033620.Google Scholar
[14] Tarruell, L., Greif, D., Uehlinger, T., Jotzu, G., and Esslinger, T. 2012. Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice. Nature, 483, 302.Google Scholar
[15] Aidelsburger, M., Lohse, M., Schweizer, C., Atala, M., Barreiro, J. T., Nascimbene, S., Cooper, N. R., Bloch, I., and Goldman, N. 2015. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nature Physics, 11, 162.Google Scholar
[16] Dauphin, A., and Goldman, N. 2013. Extracting the Chern number from the dynamics of a Fermi gas: implementing a quantum Hall bar for cold atoms. Phys. Rev. Lett., 111, 135302.Google Scholar
[17] Cohen-Tannoudji, C., Diu, B., and Lalöe, F. 1991. Quantum Mechanics. New York: Wiley.
[18] Harper, P. G. 1955. Single band motion of conduction electrons in a uniform magnetic field. Proc. Phys. Soc. A, 68, 879.Google Scholar
[19] Aharonov, Y., and Bohm, D. 1959. Significance of electromagnetic potentials in quantum theory. Phys. Rev., 115, 485.Google Scholar
[20] Jaksch, D., and Zoller, P. 2003. Creation of effective magnetic fields in optical lattices: the Hofstadter butterfly for cold neutral atoms. New Journal of Physics, 5, 56.Google Scholar
[21] Gerbier, F., and Dalibard, J. 2010. Gauge fields for ultracold atoms in optical superlattices. New Journal of Physics, 12, 3007.Google Scholar
[22] Ruostekoski, J., Dunne, G. V, and Javanainen, J. 2002. Particle number fractionalization of an atomic Fermi-Dirac gas in an optical lattice. Phys. Rev. Lett., 88, 180401.Google Scholar
[23] Mueller, E. 2004. Artificial electromagnetism for neutral atoms: Escher staircase and Laughlin liquids. Phys. Rev. A, 70, 041603.Google Scholar
[24] Alba, E., Fernandez-Gonzalvo, X., Mur-Petit, J., Pachos, J., and Garcia-Ripoll, J. 2011. Seeing topological order in time-of-flight measurements. Phys. Rev. Lett., 107, 235301.Google Scholar
[25] Goldman, N., Anisimovas, E., Gerbier, F., Öhberg, P., Spielman, I. B., and Juzeliunas, G. 2013. Measuring topology in a laser-coupled honeycomb lattice: from Chern insulators to topological semi-metals. New Journal of Physics, 15, 3025.Google Scholar
[26] Anisimovas, E., Gerbier, F., Andrijauskas, T., and Goldman, N. 2014. Design of lasercoupled honeycomb optical lattices supporting Chern insulators. Phys. Rev. A, 89, 013632.Google Scholar
[27] Celi, A., Massignan, P., Ruseckas, J., Goldman, N., Spielman, I. B., Juzeliunas, G., and Lewenstein, M. 2014. Synthetic gauge fields in synthetic dimensions. Phys. Rev. Lett., 112, 043001.Google Scholar
[28] Mancini, M., Pagano, G., Cappellini, G., Livi, L., Rider, M., Catani, J., Sias, C., Zoller, P., Inguscio, M., Dalmonte, M., and Fallani, L. 2015. Observation of chiral edge states with neutral fermions in synthetic Hall ribbons. Science 349, 1510.Google Scholar
[29] Stuhl, B. K., Lu, H. I., Aycock, L. M., Genkina, D., and Spielman, I. B. 2015. Visualizing edge states with an atomic Bose gas in the quantum Hall regime. Science 349, 1514.Google Scholar
[30] Eckardt, A., and Holthaus, M. 2007. AC-induced superfluidity. Europhysics Letters (EPL), 80, 50004.Google Scholar
[31] Lim, L.-K., Hemmerich, A., and Morais Smith, C. 2010. Artificial staggered magnetic field for ultracold atoms in optical lattices. Phys. Rev. A, 81, 023404.Google Scholar
[32] Bermudez, A., Schaetz, T., and Porras, D. 2011. Synthetic gauge fields for vibrational excitations of trapped ions. Phys. Rev. Lett., 107, 150501.Google Scholar
[33] Hauke, P., Tieleman, O., Celi, A., Ölschläger, C., Simonet, J., Struck, J., Weinberg, M., Windpassinger, P., Sengstock, K., Lewenstein, M., and Eckardt, A. 2012. Non- Abelian gauge fields and topological insulators in shaken optical lattices. Phys. Rev. Lett., 109, 145301.Google Scholar
[34] Goldman, N., Dalibard, J., Aidelsburger, M., and Cooper, N. R. 2015. Periodicallydriven quantum matter: the case of resonant modulations. Phys. Rev. A 91, 033632.Google Scholar
[35] Aidelsburger, M., Atala, M., Nascimbene, S., Trotzky, S., Chen, Y.-A., and Bloch, I. 2011. Experimental realization of strong effective magnetic fields in an optical lattice. Phys. Rev. Lett., 107, 255301.Google Scholar
[36] Aidelsburger, M., Atala, M., Lohse, M., Barreiro, J. T., Paredes, B., and Bloch, I. 2013. Realization of the Hofstadter Hamiltonian with ultracold atoms in optical lattices. Phys. Rev. Lett., 111, 85301.Google Scholar
[37] Miyake, H., Siviloglou, G. A., Kennedy, C. J., Burton, W. C., and Ketterle, W. 2013. Realizing the Harper Hamiltonian with laser-assisted tunneling in optical lattices. Phys. Phys. Lett., 111, 185302.Google Scholar
[38] Goldman, N., and Dalibard, J. 2014. Periodically-driven quantum systems: effective Hamiltonians and engineered gauge fields. Phys. Rev. X., 4, 031027.Google Scholar
[39] Rahav, S., Gilary, I., and Fishman, S. 2003. Effective Hamiltonians for periodically driven systems. Phys. Rev. A, 68, 013820.Google Scholar
[40] Bukov, M., D'Alessio, L., and Polkovnikov, A. 2015. Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering. Adv. Phys. 64, 139.Google Scholar
[41] Kolovsky, A. R. 2011. Creating artificial magnetic fields for cold atoms by photonassisted tunneling. Europhysics Letters (EPL), 93, 20003.Google Scholar
[42] Creffield, C. E., and Sols, F. 2013. Comment on “Creating artificial magnetic fields for cold atoms by photon-assisted tunneling” by Kolovsky, A. R. Europhysics Letters (EPL), 101, 40001.Google Scholar
[43] Baur, S. K., Schleier-Smith, M. H., and Cooper, N. R. 2014. Dynamic optical superlattices with topological bands. Phys. Rev. A 89, 051605(R).Google Scholar
[44] Arimondo, E., Ciampinia, D., Eckardt, A., Holthause, M., and Morsch, O. 2012. Kilohertz-driven Bose–Einstein condensates in optical lattices. Adv. At. Molec. Opt. Phys., 61, 515.Google Scholar
[45] Struck, J., Ölschläger, C., Weinberg, M., Hauke, P., Simonet, J., Eckardt, A., Lewenstein, M., Sengstock, K., and Windpassinger, P. 2012. Tunable gauge potential for neutral and spinless particles in driven optical lattices. Phys. Rev. Lett., 108, 225304.Google Scholar
[46] Struck, J., Ölschläger, C., Le Targat, R., Soltan-Panahi, P., Eckardt, A., Lewenstein, M., Windpassinger, P., and Sengstock, K. 2011. Quantum simulation of frustrated classical magnetism in triangular optical lattices. Science, 333, 996.Google Scholar
[47] Struck, J., Weinberg, M., Ölschläger, C., Windpassinger, P., Simonet, J., Sengstock, K., Höppner, R., Hauke, P., Eckardt, A., Lewenstein, M., and Mathey, L. 2013. Engineering ising-XY spin-models in a triangular lattice using tunable artificial gauge fields. Nature Physics, 9, 738.Google Scholar
[48] Jotzu, G., Messer, M., Desbuquois, R., Lebrat, M., Uehlinger, T., Greif, D., and Esslinger, T. 2014. Experimental realisation of the topological Haldane model. Nature, 515, 237.Google Scholar
[49] Fetter, A. L. 2009. Rotating trapped Bose-Einstein condensates. Rev. Mod. Phys., 81, 647.Google Scholar
[50] Madison, K. W., Chevy, F., Wohlleben, W., and Dalibard, J. 2000. Vortex formation in a stirred Bose-Einstein condensate. Phys. Rev. Lett., 84, 806.Google Scholar
[51] Abo-Shaeer, J. R., Raman, C., Vogels, J. M. and Ketterle, W. 2001. Observation of vortex lattices in Bose-Einstein condensates. Science, 292, 476.Google Scholar
[52] Hodby, E., Hechenblaikner, G., Hopkins, S. A., Marago, O. M., and Foot, C. J. 2002. Vortex nucleation in Bose-Einstein condensates in an oblate, purely magnetic potential. Phys. Rev. Lett., 88, 010405.Google Scholar
[53] Coddington, I., Haljan, P. C., Engels, P., Schweikhard, V., Tung, S., and Cornell, E. A. 2004. Experimental studies of equilibrium vortex properties in a Bose-condensed gas. Phys. Rev. A, 70, 063607.Google Scholar
[54] Schweikhard, V., Coddington, I., Engels, P., Mogendorff, V. P., and Cornell, E. A. 2004. Rapidly rotating Bose-Einstein condensates in and near the lowest Landau level. Phys. Rev. Lett., 92, 040404.Google Scholar
[55] Tung, S., Schweikhard, V., and Cornell, E. A. 2006. Observation of vortex pinning in Bose-Einstein condensates. Phys. Rev. Lett., 97, 240402.Google Scholar
[56] Hemmerich, A., and Morais Smith, C. 2007. Excitation of a d-density wave in an optical lattice with driven tunneling. Phys. Rev. Lett., 99, 113002.Google Scholar
[57] Williams, R. A., Al-Assam, S., and Foot, C. J. 2010. Observation of vortex nucleation in a rotating two-dimensional lattice of Bose-Einstein condensates. Phys. Rev. Lett., 104, 050404.Google Scholar
[58] Berry, M. V. 1984. Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. London A, 392, 45.Google Scholar
[59] Dum, R., and Olshanii, M. 1996. Gauge structures in atom-laser interaction: Bloch oscillations in a dark lattice. Phys. Rev. Lett., 76, 1788.Google Scholar
[60] Lin, Y.-J., Compton, R. L., Jiménez-Garcıa, K., Porto, J. V., and Spielman, I. B. 2009. Synthetic magnetic fields for ultracold neutral atoms. Nature, 462, 628.Google Scholar
[61] Cooper, N. R. 2011. Optical flux lattices for ultracold atomic gases. Phys. Rev. Lett., 106.Google Scholar
[62] Cooper, N. R., and Moessner, R. 2012. Designing topological bands in reciprocal space. Phys. Rev. Lett., 109, 215302.Google Scholar
[63] Cooper, N. R., and Dalibard, J. 2013. Reaching fractional quantum Hall states with optical flux lattices. Phys. Rev. Lett., 110, 185301.Google Scholar
[64] Cooper, N. R., and Dalibard, J. 2011. Optical flux lattices for two-photon dressed states. Europhysics Letters, 95, 66004.Google Scholar
[65] LeBlanc, L. J., Jiménez-Garcıa, K., Williams, R. A., Beeler, M. C., Perry, A. R., Phillips, W. D., and Spielman, I. B. 2012. Observation of a superfluid Hall effect. Proc. Natl. Acad. Sci. (PNAS), 109, 10811.Google Scholar
[66] Abanin, D. A., Kitagawa, T., Bloch, I., and Demler, E. 2013. Interferometric approach to measuring band topology in 2D optical lattices. Phys. Rev. Lett., 110, 165304.Google Scholar
[67] Duca, L., Li, T., Reitter, M., Bloch, I., Schleier-Smith, M., and Schneider, U. 2015. An Aharonov-Bohm interferometer for determining Bloch band topology. Science, 347, 288.Google Scholar
[68] Liu, Xiong-Jun, Law, K. T., Ng, T. K., and Lee Patrick, A. 2013. Detecting topological phases in cold atoms. Phys. Rev. Lett., 111, 120402.Google Scholar
[69] Liu, Xiong-Jun, Liu, Xin, Wu, Congjun, and Sinova, Jairo. 2010. Quantum anomalous Hall effect with cold atoms trapped in a square lattice. Phys. Rev. A, 81, 033622.Google Scholar
[70] Stanescu, T. D., Galitski, V., and Das Sarma, S. 2010. Topological states in twodimensional optical lattices. Phys. Rev. A, 82, 013608.Google Scholar
[71] Goldman, N., Beugnon, J., and Gerbier, F. 2012. Detecting chiral edge states in the Hofstadter optical lattice. Phys. Rev. Lett., 108, 255303.Google Scholar
[72] Buchhold, M., Cocks, D., and Hofstetter, W. 2012. Effects of smooth boundaries on topological edge modes in optical lattices. Phys. Rev. A, 85, 63614.Google Scholar
[73] Goldman, N., Dalibard, J., Dauphin, A., Gerbier, F., Lewenstein, M., Zoller, P., and Spielman, I. B. 2013. Direct imaging of topological edge states in cold-atom systems. Proc. Natl. Acad. Sci. (PNAS), 110, 1.Google Scholar
[74] Hasan, M. Z., and Kane, C. L. 2010. Colloquium: topological insulators. Rev. Mod. Phys., 82, 3045.Google Scholar
[75] Qi, X.-L., and Zhang, S.-C. 2011. Topological insulators and superconductors. Rev. Mod. Phys., 83, 1057.Google Scholar
[76] Goldman, N., Satija, I., Nikolic, P., Bermudez, A., Martin-Delgado, M. A., Lewenstein, M., and Spielman, I. B. 2010. Realistic time-reversal invariant topological insulators with neutral atoms. Phys. Rev. Lett., 105, 255302.Google Scholar
[77] Bermudez, A., Mazza, L., Rizzi, M., Goldman, N., Lewenstein, M., and Martin- Delgado, M. A. 2010. Wilson fermions and axion electrodynamics in optical lattices. Phys. Rev. Lett., 105, 190404.Google Scholar
[78] Béri, B., and Cooper, N. R. 2011. Z2 Topological insulators in ultracold atomic gases. Phys. Rev. Lett., 107, 145301.Google Scholar
[79] Essin, A. M., and Gurarie, V. 2012. Antiferromagnetic topological insulators in cold atomic gases. Phys. Rev. B, 85, 195116.Google Scholar
[80] Sorensen, A. S., Demler, E., and Lukin, M. D. 2005. Fractional quantum Hall states of atoms in optical lattices. Phys. Rev. Lett., 94, 086803.Google Scholar
[81] Palmer, R. N., and Jaksch, D. 2006. High-field fractional quantum Hall effect in optical lattices. Phys. Rev. Lett., 96, 180407.Google Scholar
[82] Möller, G., and Cooper, N. R. 2009. Composite fermion theory for bosonic quantum Hall states on lattices. Phys. Rev. Lett., 103, 105303.Google Scholar
[83] Sterdyniak, A., Regnault, N., and Möller, G. 2012. Particle entanglement spectra for quantum Hall states on lattices. Phys. Rev. B, 86, 165314.Google Scholar
[84] Hormozi, L., Möller, G., and Simon, S. H. 2012. Fractional quantum Hall effect of lattice bosons near commensurate flux. Phys. Rev. Lett., 108, 256809.Google Scholar
[85] Sterdyniak, A., Bernevig, B. Andrei, Cooper, Nigel, R., and Regnault, N. 2015. Interacting bosons in topological optical flux lattices. Phys. Rev. B, 91, 035115.Google Scholar
[86] Baur, S. K., and Cooper, N. R. 2012. Coupled ferromagnetic and nematic ordering of fermions in an optical flux lattice. Phys. Rev. Lett., 109, 265301.Google Scholar
[87] Yao, N. Y., Gorshkov, A. V., Laumann, C. R., Luchli, A. M., Ye, J., and Lukin, M. D. 2013. Realizing fractional Chern insulators in dipolar spin systems. Phys. Rev. Lett., 110, 185302.Google Scholar

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