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Käming N, Dawid A, Kottmann K, Lewenstein M, Sengstock K, Dauphin A, Weitenberg C. Unsupervised machine learning of topological phase transitions from experimental data. MACHINE LEARNING: SCIENCE AND TECHNOLOGY 2021. [DOI: 10.1088/2632-2153/abffe7] [Citation(s) in RCA: 16] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/04/2023] Open
Abstract
Abstract
Identifying phase transitions is one of the key challenges in quantum many-body physics. Recently, machine learning methods have been shown to be an alternative way of localising phase boundaries from noisy and imperfect data without the knowledge of the order parameter. Here, we apply different unsupervised machine learning techniques, including anomaly detection and influence functions, to experimental data from ultracold atoms. In this way, we obtain the topological phase diagram of the Haldane model in a completely unbiased fashion. We show that these methods can successfully be applied to experimental data at finite temperatures and to the data of Floquet systems when post-processing the data to a single micromotion phase. Our work provides a benchmark for the unsupervised detection of new exotic phases in complex many-body systems.
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Stepanenko AA, Gorlach MA. Probing the topology of the two-photon bands via time-dependent quantum walks. OPTICS LETTERS 2021; 46:2726-2729. [PMID: 34061098 DOI: 10.1364/ol.425841] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/24/2021] [Accepted: 05/10/2021] [Indexed: 06/12/2023]
Abstract
Topological protection of quantum correlations opens new horizons and opportunities in quantum technologies. A variety of topological effects has recently been observed in qubit networks. However, the experimental identification of the topological phase still remains challenging, especially in the entangled many-body case. Here, we propose an approach to independently probe single- and two-photon topological invariants from the time evolution of the two-photon state in a one-dimensional array of qubits. Extending the bulk-boundary correspondence to the two-photon scenario, we show that an appropriate choice of the initial state enables the retrieval of the topological invariant for the different types of the two-photon states in the interacting Su-Schrieffer-Heeger model. Our analysis of the Zak phase reveals additional facets of topological protection in the case of collapse of bound photon pairs.
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Julià-Farré S, Müller M, Lewenstein M, Dauphin A. Self-Trapped Polarons and Topological Defects in a Topological Mott Insulator. PHYSICAL REVIEW LETTERS 2020; 125:240601. [PMID: 33412044 DOI: 10.1103/physrevlett.125.240601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2020] [Accepted: 11/05/2020] [Indexed: 06/12/2023]
Abstract
Many-body interactions in topological quantum systems can give rise to new phases of matter, which simultaneously exhibit both rich spatial features and topological properties. In this work, we consider spinless fermions on a checkerboard lattice with nearest and next-to-nearest neighbor interactions. We calculate the phase diagram at half filling, which presents, in particular, an interaction-induced quantum anomalous Hall phase. We study the system at incommensurate fillings using an unrestricted Hartree-Fock ansatz and report a rich zoo of solutions such as self-trapped polarons and domain walls above an interaction-induced topological insulator. We find that, as a consequence of the interplay between the interaction-induced topology and topological defects, these domain walls separate two phases with opposite topological invariants and host topologically protected chiral edge states. Finally, we discuss experimental prospects to observe these novel phenomena in a quantum simulator based on laser-dressed Rydberg atoms in an optical lattice.
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Affiliation(s)
- Sergi Julià-Farré
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Avinguda Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
| | - Markus Müller
- Institute for Quantum Information, RWTH Aachen University, D-52074 Aachen, Germany
- Peter Grünberg Institute, Theoretical Nanoelectronics, Forschungszentrum Jülich, D-52428 Jülich, Germany
| | - Maciej Lewenstein
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Avinguda Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
- ICREA, Passeig Lluís Companys 23, 08010 Barcelona, Spain
| | - Alexandre Dauphin
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Avinguda Carl Friedrich Gauss 3, 08860 Castelldefels (Barcelona), Spain
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Motruk J, Na I. Detecting Fractional Chern Insulators in Optical Lattices through Quantized Displacement. PHYSICAL REVIEW LETTERS 2020; 125:236401. [PMID: 33337233 DOI: 10.1103/physrevlett.125.236401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/02/2020] [Accepted: 10/22/2020] [Indexed: 06/12/2023]
Abstract
The realization of interacting topological states of matter such as fractional Chern insulators (FCIs) in cold atom systems has recently come within experimental reach due to the engineering of optical lattices with synthetic gauge fields providing the required topological band structures. However, detecting their occurrence might prove difficult since transport measurements akin to those in solid state systems are challenging to perform in cold atom setups and alternatives have to be found. We show that for a ν=1/2 FCI state realized in the lowest band of a Harper-Hofstadter model of interacting bosons confined by a harmonic trapping potential, the fractionally quantized Hall conductivity σ_{xy} can be accurately determined by the displacement of the atomic cloud under the action of a constant force which provides a suitable experimentally measurable signal for detecting the topological nature of the state. Using matrix-product state algorithms, we show that, in both cylinder and square geometries, the movement of the particle cloud in time under the application of a constant force field on top of the confining potential is proportional to σ_{xy} for an extended range of field strengths.
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Affiliation(s)
- Johannes Motruk
- Department of Physics, University of California, Berkeley, California 94720, USA
- Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - Ilyoun Na
- Department of Physics, University of California, Berkeley, California 94720, USA
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Ulčakar L, Mravlje J, Rejec T. Kibble-Zurek Behavior in Disordered Chern Insulators. PHYSICAL REVIEW LETTERS 2020; 125:216601. [PMID: 33274996 DOI: 10.1103/physrevlett.125.216601] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/13/2020] [Accepted: 10/29/2020] [Indexed: 06/12/2023]
Abstract
Even though no local order parameter in the sense of the Landau theory exists for topological quantum phase transitions in Chern insulators, the highly nonlocal Berry curvature exhibits critical behavior near a quantum critical point. We investigate the critical properties of its real space analog, the local Chern marker, in weakly disordered Chern insulators. Because of disorder, inhomogeneities appear in the spatial distribution of the local Chern marker. Their size exhibits power-law scaling with the critical exponent matching the one extracted from the Berry curvature of a clean system. We drive the system slowly through such a quantum phase transition. The characteristic size of inhomogeneities in the nonequilibrium postquench state obeys the Kibble-Zurek scaling. In this setting, the local Chern marker thus does behave in a similar way as a local order parameter for a symmetry breaking second order phase transition. The Kibble-Zurek scaling also holds for the inhomogeneities in the spatial distribution of excitations and of the orbital polarization.
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Affiliation(s)
- Lara Ulčakar
- Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia and Faculty for Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - Jernej Mravlje
- Jozef Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
| | - Tomaž Rejec
- Jozef Stefan Institute, Jamova 39, Ljubljana, Slovenia and Faculty for Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
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6
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Higashikawa S, Nakagawa M, Ueda M. Floquet Chiral Magnetic Effect. PHYSICAL REVIEW LETTERS 2019; 123:066403. [PMID: 31491132 DOI: 10.1103/physrevlett.123.066403] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/19/2018] [Revised: 04/24/2019] [Indexed: 06/10/2023]
Abstract
A single Weyl fermion, which is prohibited in static lattice systems by the Nielsen-Ninomiya theorem, is shown to be realized in a periodically driven three-dimensional lattice system with a topologically nontrivial Floquet unitary operator, manifesting the chiral magnetic effect. We give a topological classification of Floquet unitary operators in the Altland-Zirnbauer symmetry classes for all dimensions, and use it to predict that all gapless surface states of topological insulators and superconductors can emerge in bulk quasienergy spectra of Floquet systems.
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Affiliation(s)
- Sho Higashikawa
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Masaya Nakagawa
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
| | - Masahito Ueda
- Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
- RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
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7
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Cooper NR, Dalibard J, Spielman IB. Topological bands for ultracold atoms. REVIEWS OF MODERN PHYSICS 2019; 91:10.1103/revmodphys.91.015005. [PMID: 32189812 PMCID: PMC7079706 DOI: 10.1103/revmodphys.91.015005] [Citation(s) in RCA: 113] [Impact Index Per Article: 22.6] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/22/2023]
Abstract
There have been significant recent advances in realizing band structures with geometrical and topological features in experiments on cold atomic gases. This review summarizes these developments, beginning with a summary of the key concepts of geometry and topology for Bloch bands. Descriptions are given of the different methods that have been used to generate these novel band structures for cold atoms and of the physical observables that have allowed their characterization. The focus is on the physical principles that underlie the different experimental approaches, providing a conceptual framework within which to view these developments. Also described is how specific experimental implementations can influence physical properties. Moving beyond single-particle effects, descriptions are given of the forms of interparticle interactions that emerge when atoms are subjected to these energy bands and of some of the many-body phases that may be sought in future experiments.
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Affiliation(s)
- N R Cooper
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - J Dalibard
- Laboratoire Kastler Brossel, Collège de France, CNRS, ENS-Université PSL, Sorbonne Université, 11 place Marcelin Berthelot, 75005, Paris, France
| | - I B Spielman
- Joint Quantum Institute, National Institute of Standards and Technology and University of Maryland, Gaithersburg, Maryland 20899, USA
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Generalized Haldane models on laser-coupling optical lattices. Sci Rep 2018; 8:12898. [PMID: 30150700 PMCID: PMC6110748 DOI: 10.1038/s41598-018-30503-9] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/18/2018] [Accepted: 07/25/2018] [Indexed: 11/28/2022] Open
Abstract
We propose two generalized Haldane models on laser-coupling optical lattices. Laser-assisted nearest neighbour tunnelings generate artificial staggered magnetic flux, facilitating the realization of topological nontrivial band structures. As generalizations of Haldane model, these models support topological insulator and semimetal phases featuring high Chern numbers. We show simple rules for computing Chern numbers of our models and display the phase diagrams. Moreover, numerical calculations of energy spectra are in perfect agreement with our theoretical expectations. Our models may serve as two new family members for generalizing Haldane model on optical lattices.
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Xu XY, Wang QQ, Pan WW, Sun K, Xu JS, Chen G, Tang JS, Gong M, Han YJ, Li CF, Guo GC. Measuring the Winding Number in a Large-Scale Chiral Quantum Walk. PHYSICAL REVIEW LETTERS 2018; 120:260501. [PMID: 30004718 DOI: 10.1103/physrevlett.120.260501] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/27/2017] [Revised: 04/25/2018] [Indexed: 05/27/2023]
Abstract
We report the experimental measurement of the winding number in an unitary chiral quantum walk. Fundamentally, the spin-orbit coupling in discrete time quantum walks is implemented via a birefringent crystal collinearly cut based on a time-multiplexing scheme. Our protocol is compact and avoids extra loss, making it suitable for realizing genuine single-photon quantum walks at a large scale. By adopting a heralded single photon as the walker and with a high time resolution technology in single-photon detection, we carry out a 50-step Hadamard discrete-time quantum walk with high fidelity up to 0.948±0.007. Particularly, we can reconstruct the complete wave function of the walker that starts the walk in a single lattice site through the local tomography of each site. Through a Fourier transform, the wave function in quasimomentum space can be obtained. With this ability, we propose and report a method to reconstruct the eigenvectors of the system Hamiltonian in quasimomentum space and directly read out the winding numbers in different topological phases (trivial and nontrivial) in the presence of chiral symmetry. By introducing nonequivalent time frames, we show that whole topological phases in a periodically driven system can also be characterized by two different winding numbers. Our method can also be extended to the high winding number situation.
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Affiliation(s)
- Xiao-Ye Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Qin-Qin Wang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Wei-Wei Pan
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Kai Sun
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Jin-Shi Xu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Geng Chen
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Jian-Shun Tang
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Ming Gong
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Yong-Jian Han
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Chuan-Feng Li
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
| | - Guang-Can Guo
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, People's Republic of China
- Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, People's Republic of China
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Tran DT, Dauphin A, Grushin AG, Zoller P, Goldman N. Probing topology by "heating": Quantized circular dichroism in ultracold atoms. SCIENCE ADVANCES 2017; 3:e1701207. [PMID: 28835930 PMCID: PMC5562418 DOI: 10.1126/sciadv.1701207] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2017] [Accepted: 07/16/2017] [Indexed: 05/30/2023]
Abstract
We reveal an intriguing manifestation of topology, which appears in the depletion rate of topological states of matter in response to an external drive. This phenomenon is presented by analyzing the response of a generic two-dimensional (2D) Chern insulator subjected to a circular time-periodic perturbation. Because of the system's chiral nature, the depletion rate is shown to depend on the orientation of the circular shake; taking the difference between the rates obtained from two opposite orientations of the drive, and integrating over a proper drive-frequency range, provides a direct measure of the topological Chern number (ν) of the populated band: This "differential integrated rate" is directly related to the strength of the driving field through the quantized coefficient η0 = ν/ℏ2, where h = 2π ℏ is Planck's constant. Contrary to the integer quantum Hall effect, this quantized response is found to be nonlinear with respect to the strength of the driving field, and it explicitly involves interband transitions. We investigate the possibility of probing this phenomenon in ultracold gases and highlight the crucial role played by edge states in this effect. We extend our results to 3D lattices, establishing a link between depletion rates and the nonlinear photogalvanic effect predicted for Weyl semimetals. The quantized circular dichroism revealed in this work designates depletion rate measurements as a universal probe for topological order in quantum matter.
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Affiliation(s)
- Duc Thanh Tran
- Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium
| | - Alexandre Dauphin
- ICFO–Institut de Ciencies Fotoniques, Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
| | - Adolfo G. Grushin
- Department of Physics, University of California, Berkeley, CA 94720, USA
- Institut Néel, CNRS and Université Grenoble Alpes, F-38042 Grenoble, France
| | - Peter Zoller
- International Solvay Institutes, Université Libre de Bruxelles, Campus Plaine, B-1050 Brussels, Belgium
- Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria
| | - Nathan Goldman
- Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium
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Yu J, Xu ZF, Lü R, You L. Dynamical Generation of Topological Magnetic Lattices for Ultracold Atoms. PHYSICAL REVIEW LETTERS 2016; 116:143003. [PMID: 27104703 DOI: 10.1103/physrevlett.116.143003] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/17/2015] [Indexed: 06/05/2023]
Abstract
We propose a scheme to dynamically synthesize a space-periodic effective magnetic field for neutral atoms by time-periodic magnetic field pulses. When atomic spin adiabatically follows the direction of the effective magnetic field, an adiabatic scalar potential together with a geometric vector potential emerges for the atomic center-of-mass motion, due to the Berry phase effect. While atoms hop between honeycomb lattice sites formed by the minima of the adiabatic potential, complex Peierls phase factors in the hopping coefficients are induced by the vector potential, and these phase factors facilitate a topological Chern insulator. With further tuning of external parameters, both a topological phase transition and topological flat bands can be achieved, highlighting realistic prospects for studying strongly correlated phenomena in this system. Our Letter presents an alternative pathway towards creating and manipulating topological states of ultracold atoms by magnetic fields.
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Affiliation(s)
- Jinlong Yu
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Zhi-Fang Xu
- MOE Key Laboratory of Fundamental Physical Quantities Measurements, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Rong Lü
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
| | - Li You
- State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing 100084, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
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12
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Price HM, Zilberberg O, Ozawa T, Carusotto I, Goldman N. Four-Dimensional Quantum Hall Effect with Ultracold Atoms. PHYSICAL REVIEW LETTERS 2015; 115:195303. [PMID: 26588394 DOI: 10.1103/physrevlett.115.195303] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/21/2015] [Indexed: 06/05/2023]
Abstract
We propose a realistic scheme to detect the 4D quantum Hall effect using ultracold atoms. Based on contemporary technology, motion along a synthetic fourth dimension can be accomplished through controlled transitions between internal states of atoms arranged in a 3D optical lattice. From a semiclassical analysis, we identify the linear and nonlinear quantized current responses of our 4D model, relating these to the topology of the Bloch bands. We then propose experimental protocols, based on current or center-of-mass-drift measurements, to extract the topological second Chern number. Our proposal sets the stage for the exploration of novel topological phases in higher dimensions.
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Affiliation(s)
- H M Price
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
| | - O Zilberberg
- Institute for Theoretical Physics, ETH Zurich, 8093 Zürich, Switzerland
| | - T Ozawa
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
| | - I Carusotto
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
| | - N Goldman
- CENOLI, Faculté des Sciences, Université Libre de Bruxelles (U.L.B.), B-1050 Brussels, Belgium
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13
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Zeuner JM, Rechtsman MC, Plotnik Y, Lumer Y, Nolte S, Rudner MS, Segev M, Szameit A. Observation of a Topological Transition in the Bulk of a Non-Hermitian System. PHYSICAL REVIEW LETTERS 2015; 115:040402. [PMID: 26252670 DOI: 10.1103/physrevlett.115.040402] [Citation(s) in RCA: 146] [Impact Index Per Article: 16.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/31/2014] [Indexed: 05/05/2023]
Abstract
We present the first experimental observation of a topological transition in a non-Hermitian system. In contrast to standard methods for examining topological properties, which involve probing edge (or surface) states, we monitor the topological transition by employing bulk dynamics only. The system is composed of a lattice of evanescently coupled optical waveguides, and non-Hermitian behavior is engineered by inducing bending loss by spatially "wiggling" every second waveguide.
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Affiliation(s)
- Julia M Zeuner
- Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
| | - Mikael C Rechtsman
- Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 16802, USA
| | - Yonatan Plotnik
- Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
| | - Yaakov Lumer
- Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
| | - Stefan Nolte
- Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
| | - Mark S Rudner
- The Niels Bohr International Academy, Niels Bohr Institute, 2100 Copenhagen, Denmark
| | - Mordechai Segev
- Department of Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
| | - Alexander Szameit
- Institute of Applied Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
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14
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Kitamura S, Tsuji N, Aoki H. Interaction-Driven Topological Insulator in Fermionic Cold Atoms on an Optical Lattice: A Design with a Density Functional Formalism. PHYSICAL REVIEW LETTERS 2015; 115:045304. [PMID: 26252693 DOI: 10.1103/physrevlett.115.045304] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/24/2014] [Indexed: 06/04/2023]
Abstract
We design an interaction-driven topological insulator for fermionic cold atoms in an optical lattice; that is, we pose the question of whether we can realize in a continuous space a spontaneous symmetry breaking induced by the interatom interaction into a topological Chern insulator. Such a state, sometimes called a "topological Mott insulator," has yet to be realized in solid-state systems, since this requires, in the tight-binding model, large off-site interactions on top of a small on-site interaction. Here, we overcome the difficulty by introducing a spin-dependent potential, where a spin-selective occupation of fermions in A and B sublattices makes the on-site interaction Pauli forbidden, while a sizeable intersite interaction is achieved by a shallow optical potential with a large overlap between neighboring Wannier orbitals. This puts the system away from the tight-binding model, so that we adopt density functional theory for cold atoms, here extended to accommodate noncollinear spin structures emerging in the topological regime, to quantitatively demonstrate the phase transition to the topological Mott insulator.
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Affiliation(s)
- Sota Kitamura
- Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
| | - Naoto Tsuji
- Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
| | - Hideo Aoki
- Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan
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15
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Duca L, Li T, Reitter M, Bloch I, Schleier-Smith M, Schneider U. An Aharonov-Bohm interferometer for determining Bloch band topology. Science 2014; 347:288-92. [DOI: 10.1126/science.1259052] [Citation(s) in RCA: 185] [Impact Index Per Article: 18.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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16
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Experimental realization of the topological Haldane model with ultracold fermions. Nature 2014; 515:237-40. [PMID: 25391960 DOI: 10.1038/nature13915] [Citation(s) in RCA: 370] [Impact Index Per Article: 37.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/19/2014] [Accepted: 09/29/2014] [Indexed: 11/08/2022]
Abstract
The Haldane model on a honeycomb lattice is a paradigmatic example of a Hamiltonian featuring topologically distinct phases of matter. It describes a mechanism through which a quantum Hall effect can appear as an intrinsic property of a band structure, rather than being caused by an external magnetic field. Although physical implementation has been considered unlikely, the Haldane model has provided the conceptual basis for theoretical and experimental research exploring topological insulators and superconductors. Here we report the experimental realization of the Haldane model and the characterization of its topological band structure, using ultracold fermionic atoms in a periodically modulated optical honeycomb lattice. The Haldane model is based on breaking both time-reversal symmetry and inversion symmetry. To break time-reversal symmetry, we introduce complex next-nearest-neighbour tunnelling terms, which we induce through circular modulation of the lattice position. To break inversion symmetry, we create an energy offset between neighbouring sites. Breaking either of these symmetries opens a gap in the band structure, which we probe using momentum-resolved interband transitions. We explore the resulting Berry curvatures, which characterize the topology of the lowest band, by applying a constant force to the atoms and find orthogonal drifts analogous to a Hall current. The competition between the two broken symmetries gives rise to a transition between topologically distinct regimes. By identifying the vanishing gap at a single Dirac point, we map out this transition line experimentally and quantitatively compare it to calculations using Floquet theory without free parameters. We verify that our approach, which allows us to tune the topological properties dynamically, is suitable even for interacting fermionic systems. Furthermore, we propose a direct extension to realize spin-dependent topological Hamiltonians.
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Goldman N, Juzeliūnas G, Öhberg P, Spielman IB. Light-induced gauge fields for ultracold atoms. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2014; 77:126401. [PMID: 25422950 DOI: 10.1088/0034-4885/77/12/126401] [Citation(s) in RCA: 173] [Impact Index Per Article: 17.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Gauge fields are central in our modern understanding of physics at all scales. At the highest energy scales known, the microscopic universe is governed by particles interacting with each other through the exchange of gauge bosons. At the largest length scales, our Universe is ruled by gravity, whose gauge structure suggests the existence of a particle-the graviton-that mediates the gravitational force. At the mesoscopic scale, solid-state systems are subjected to gauge fields of different nature: materials can be immersed in external electromagnetic fields, but they can also feature emerging gauge fields in their low-energy description. In this review, we focus on another kind of gauge field: those engineered in systems of ultracold neutral atoms. In these setups, atoms are suitably coupled to laser fields that generate effective gauge potentials in their description. Neutral atoms 'feeling' laser-induced gauge potentials can potentially mimic the behavior of an electron gas subjected to a magnetic field, but also, the interaction of elementary particles with non-Abelian gauge fields. Here, we review different realized and proposed techniques for creating gauge potentials-both Abelian and non-Abelian-in atomic systems and discuss their implication in the context of quantum simulation. While most of these setups concern the realization of background and classical gauge potentials, we conclude with more exotic proposals where these synthetic fields might be made dynamical, in view of simulating interacting gauge theories with cold atoms.
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Affiliation(s)
- N Goldman
- College de France, 11 place Marcelin Berthelot & Laboratoire Kastler Brossel, CNRS, UPMC, ENS, 24 rue Lhomond, 75005 Paris, France
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Price HM, Ozawa T, Carusotto I. Quantum mechanics with a momentum-space artificial magnetic field. PHYSICAL REVIEW LETTERS 2014; 113:190403. [PMID: 25415888 DOI: 10.1103/physrevlett.113.190403] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2014] [Indexed: 06/04/2023]
Abstract
The Berry curvature is a geometrical property of an energy band which acts as a momentum space magnetic field in the effective Hamiltonian describing single-particle quantum dynamics. We show how this perspective may be exploited to study systems directly relevant to ultracold gases and photonics. Given the exchanged roles of momentum and position, we demonstrate that the global topology of momentum space is crucially important. We propose an experiment to study the Harper-Hofstadter Hamiltonian with a harmonic trap that will illustrate the advantages of this approach and that will also constitute the first realization of magnetism on a torus.
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Affiliation(s)
- Hannah M Price
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
| | - Tomoki Ozawa
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
| | - Iacopo Carusotto
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
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Hauke P, Lewenstein M, Eckardt A. Tomography of band insulators from quench dynamics. PHYSICAL REVIEW LETTERS 2014; 113:045303. [PMID: 25105629 DOI: 10.1103/physrevlett.113.045303] [Citation(s) in RCA: 18] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/03/2014] [Indexed: 06/03/2023]
Abstract
We propose a simple scheme for tomography of band-insulating states in one- and two-dimensional optical lattices with two sublattice states. In particular, the scheme maps out the Berry curvature in the entire Brillouin zone and extracts topological invariants such as the Chern number. The measurement relies on observing--via time-of-flight imaging--the time evolution of the momentum distribution following a sudden quench in the band structure. We consider two examples of experimental relevance: the Harper model with π flux and the Haldane model on a honeycomb lattice. Moreover, we illustrate the performance of the scheme in the presence of a parabolic trap, noise, and finite measurement resolution.
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Affiliation(s)
- Philipp Hauke
- Institut für Quantenoptik und Quanteninformation, Österreichische Akademie der Wissenschaften, Technikerstraße 21A, A-6020 Innsbruck, Austria and Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 25, A-6020 Innsbruck, Austria
| | - Maciej Lewenstein
- ICFO-Institut de Ciències Fotòniques, Parc Mediterrani de la Tecnologia, E-08860 Castelldefels, Spain and ICREA-Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, E-08010 Barcelona, Spain
| | - André Eckardt
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, D-01187 Dresden, Germany
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Ozawa T, Carusotto I. Anomalous and quantum Hall effects in lossy photonic lattices. PHYSICAL REVIEW LETTERS 2014; 112:133902. [PMID: 24745421 DOI: 10.1103/physrevlett.112.133902] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/25/2013] [Indexed: 06/03/2023]
Abstract
We theoretically discuss analogues of the anomalous and the integer quantum Hall effect in driven-dissipative two-dimensional photonic lattices in the presence of a synthetic gauge field. Photons are coherently injected by a spatially localized pump, and the transverse shift of the in-plane light distribution under the effect of an additional uniform force is considered. Depending on pumping parameters, the transverse shift turns out to be proportional either to the global Chern number (integer quantum Hall effect) or to the local Berry curvature (anomalous Hall effect). This suggests a viable route to experimentally measure these quantities in photonic lattices.
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Affiliation(s)
- Tomoki Ozawa
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
| | - Iacopo Carusotto
- INO-CNR BEC Center and Dipartimento di Fisica, Università di Trento, I-38123 Povo, Italy
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