1
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Bauer NP, Budich JC, Trauzettel B, Calzona A. Quench-Probe Setup as Analyzer of Fractionalized Entanglement Spreading. PHYSICAL REVIEW LETTERS 2023; 130:190401. [PMID: 37243652 DOI: 10.1103/physrevlett.130.190401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/13/2022] [Accepted: 04/11/2023] [Indexed: 05/29/2023]
Abstract
We propose a novel spatially inhomogeneous setup for revealing quench-induced fractionalized excitations in entanglement dynamics. In this quench-probe setting, the region undergoing a quantum quench is tunnel coupled to a static region, the probe. Subsequently, the time-dependent entanglement signatures of a tunable subset of excitations propagating to the probe are monitored by energy selectivity. We exemplify the power of this generic approach by identifying a unique dynamical signature associated with the presence of an isolated Majorana zero mode in the postquench Hamiltonian. In this case excitations emitted from the topological part of the system give rise to a fractionalized jump of log(2)/2 in the entanglement entropy of the probe. This dynamical effect is highly sensitive to the localized nature of the Majorana zero mode, but does not require the preparation of a topological initial state.
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Affiliation(s)
- Nicolas P Bauer
- Institute of Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
- Würzburg-Dresden Cluster of Excellence ct.qmat, Germany
| | - Jan Carl Budich
- Würzburg-Dresden Cluster of Excellence ct.qmat, Germany
- Institute of Theoretical Physics, Technische Universität Dresden, 01062 Dresden, Germany
| | - Björn Trauzettel
- Institute of Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
- Würzburg-Dresden Cluster of Excellence ct.qmat, Germany
| | - Alessio Calzona
- Institute of Theoretical Physics and Astrophysics, University of Würzburg, 97074 Würzburg, Germany
- Würzburg-Dresden Cluster of Excellence ct.qmat, Germany
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2
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Marino J, Eckstein M, Foster MS, Rey AM. Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2022; 85:116001. [PMID: 36075190 DOI: 10.1088/1361-6633/ac906c] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/11/2022] [Accepted: 09/08/2022] [Indexed: 06/15/2023]
Abstract
We overview the concept of dynamical phase transitions (DPTs) in isolated quantum systems quenched out of equilibrium. We focus on non-equilibrium transitions characterized by an order parameter, which features qualitatively distinct temporal behavior on the two sides of a certain dynamical critical point. DPTs are currently mostly understood as long-lived prethermal phenomena in a regime where inelastic collisions are incapable to thermalize the system. The latter enables the dynamics to substain phases that explicitly break detailed balance and therefore cannot be encompassed by traditional thermodynamics. Our presentation covers both cold atoms as well as condensed matter systems. We revisit a broad plethora of platforms exhibiting pre-thermal DPTs, which become theoretically tractable in a certain limit, such as for a large number of particles, large number of order parameter components, or large spatial dimension. The systems we explore include, among others, quantum magnets with collective interactions,ϕ4quantum field theories, and Fermi-Hubbard models. A section dedicated to experimental explorations of DPTs in condensed matter and AMO systems connects this large variety of theoretical models.
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Affiliation(s)
- Jamir Marino
- Institut für Physik, Johannes Gutenberg-Universität Mainz, D-55099 Mainz, Germany
| | - Martin Eckstein
- Department of Physics, University of Erlangen-Nürnberg, 91058 Erlangen, Germany
| | - Matthew S Foster
- Department of Physics and Astronomy, Rice University, Houston, TX 77005, United States of America
- Rice Center for Quantum Materials, Rice University, Houston, TX 77005, United States of America
| | - Ana Maria Rey
- JILA, National Institute of Standards and Technology, and Department of Physics,University of Colorado, Boulder, CO 80309, United States of America
- Center for Theory of Quantum Matter, University of Colorado, Boulder, CO 80309, United States of America
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3
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Aftab T, Sabeeh K. Quantum quench of photoinduced semi-Dirac materials: Hall response. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2022; 34:425701. [PMID: 35952639 DOI: 10.1088/1361-648x/ac8904] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/20/2022] [Accepted: 08/11/2022] [Indexed: 06/15/2023]
Abstract
In this work, far from equilibrium Hall response of semi-Dirac materials is studied. This required preparing the system in non-equilibrium states through a quantum quench protocol. We show that in the non-equilibrium setting, there is non-zero Hall response even when instantaneous time reversal symmetry (TRS) is present and the Hall current persists for long times. This is in contrast to the equilibrium case where the system is required to break TRS for a Hall response. This highlights unique features of far from equilibrium response in semi-Dirac materials that are not present in the corresponding equilibrium state.
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Affiliation(s)
- Tayyaba Aftab
- Department of Physics, Allama Iqbal Open University, Islamabad 44000, Pakistan
| | - Kashif Sabeeh
- Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan
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4
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Jung PS, Pyrialakos GG, Wu FO, Parto M, Khajavikhan M, Krolikowski W, Christodoulides DN. Thermal control of the topological edge flow in nonlinear photonic lattices. Nat Commun 2022; 13:4393. [PMID: 35906224 PMCID: PMC9338248 DOI: 10.1038/s41467-022-32069-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/25/2022] [Accepted: 07/18/2022] [Indexed: 11/29/2022] Open
Abstract
The chaotic evolution resulting from the interplay between topology and nonlinearity in photonic systems generally forbids the sustainability of optical currents. Here, we systematically explore the nonlinear evolution dynamics in topological photonic lattices within the framework of optical thermodynamics. By considering an archetypical two-dimensional Haldane photonic lattice, we discover several prethermal states beyond the topological phase transition point and a stable global equilibrium response, associated with a specific optical temperature and chemical potential. Along these lines, we provide a consistent thermodynamic methodology for both controlling and maximizing the unidirectional power flow in the topological edge states. This can be achieved by either employing cross-phase interactions between two subsystems or by exploiting self-heating effects in disordered or Floquet topological lattices. Our results indicate that photonic topological systems can in fact support robust photon transport processes even under the extreme complexity introduced by nonlinearity, an important feature for contemporary topological applications in photonics. The nonlinear evolution dynamics in topological photonic lattices is systematically investigated within the framework of optical thermodynamics. This approach allows for the precise prediction of topological currents even under the extreme complexity introduced by nonlinearity.
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Affiliation(s)
- Pawel S Jung
- College of Optics & Photonics-CREOL, University of Central Florida, Orlando, FL, 32816, USA.,Faculty of Physics, Warsaw University of Technology, Koszykowa 75, 00-662, Warsaw, Poland
| | - Georgios G Pyrialakos
- College of Optics & Photonics-CREOL, University of Central Florida, Orlando, FL, 32816, USA
| | - Fan O Wu
- College of Optics & Photonics-CREOL, University of Central Florida, Orlando, FL, 32816, USA
| | - Midya Parto
- College of Optics & Photonics-CREOL, University of Central Florida, Orlando, FL, 32816, USA
| | - Mercedeh Khajavikhan
- College of Optics & Photonics-CREOL, University of Central Florida, Orlando, FL, 32816, USA.,Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA, USA
| | - Wieslaw Krolikowski
- Laser Physics Centre, Research School of Physics and Engineering, Australian National University, Canberra, ACT, 0200, Australia.,Science Program, Texas A&M University at Qatar, Doha, Qatar
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5
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Yu D, Peng B, Chen X, Liu XJ, Yuan L. Topological holographic quench dynamics in a synthetic frequency dimension. LIGHT, SCIENCE & APPLICATIONS 2021; 10:209. [PMID: 34620837 PMCID: PMC8497532 DOI: 10.1038/s41377-021-00646-y] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/07/2021] [Revised: 09/10/2021] [Accepted: 09/14/2021] [Indexed: 05/06/2023]
Abstract
The notion of topological phases extended to dynamical systems stimulates extensive studies, of which the characterization of nonequilibrium topological invariants is a central issue and usually necessitates the information of quantum dynamics in both the time and momentum dimensions. Here, we propose the topological holographic quench dynamics in synthetic dimension, and also show it provides a highly efficient scheme to characterize photonic topological phases. A pseudospin model is constructed with ring resonators in a synthetic lattice formed by frequencies of light, and the quench dynamics is induced by initializing a trivial state, which evolves under a topological Hamiltonian. Our key prediction is that the complete topological information of the Hamiltonian is encoded in quench dynamics solely in the time dimension, and is further mapped to lower-dimensional space, manifesting the holographic features of the dynamics. In particular, two fundamental time scales emerge in the dynamical evolution, with one mimicking the topological band on the momentum dimension and the other characterizing the residue time evolution of the state after the quench. For this, a universal duality between the quench dynamics and the equilibrium topological phase of the spin model is obtained in the time dimension by extracting information from the field evolution dynamics in modulated ring systems in simulations. This work also shows that the photonic synthetic frequency dimension provides an efficient and powerful way to explore the topological nonequilibrium dynamics.
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Affiliation(s)
- Danying Yu
- State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, 200240, Shanghai, China
| | - Bo Peng
- State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, 200240, Shanghai, China
| | - Xianfeng Chen
- State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, 200240, Shanghai, China
- Shanghai Research Center for Quantum Sciences, 201315, Shanghai, China
- Jinan Institute of Quantum Technology, 250101, Jinan, China
- Collaborative Innovation Center of Light Manipulations and Applications, Shandong Normal University, 250358, Jinan, China
| | - Xiong-Jun Liu
- International Center for Quantum Materials and School of Physics, Peking University, 100871, Beijing, China.
- Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology, 518055, Shenzhen, China.
| | - Luqi Yuan
- State Key Laboratory of Advanced Optical Communication Systems and Networks, School of Physics and Astronomy, Shanghai Jiao Tong University, 200240, Shanghai, China.
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6
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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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7
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Ünal FN, Bouhon A, Slager RJ. Topological Euler Class as a Dynamical Observable in Optical Lattices. PHYSICAL REVIEW LETTERS 2020; 125:053601. [PMID: 32794847 DOI: 10.1103/physrevlett.125.053601] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Revised: 06/21/2020] [Accepted: 07/06/2020] [Indexed: 05/22/2023]
Abstract
The last years have witnessed rapid progress in the topological characterization of out-of-equilibrium systems. We report on robust signatures of a new type of topology-the Euler class-in such a dynamical setting. The enigmatic invariant (ξ) falls outside conventional symmetry-eigenvalue indicated phases and, in simplest incarnation, is described by triples of bands that comprise a gapless pair featuring 2ξ stable band nodes, and a gapped band. These nodes host non-Abelian charges and can be further undone by converting their charge upon intricate braiding mechanisms, revealing that Euler class is a fragile topology. We theoretically demonstrate that quenching with nontrivial Euler Hamiltonian results in stable monopole-antimonopole pairs, which in turn induce a linking of momentum-time trajectories under the first Hopf map, making the invariant experimentally observable. Detailing explicit tomography protocols in a variety of cold-atom setups, our results provide a basis for exploring new topologies and their interplay with crystalline symmetries in optical lattices beyond paradigmatic Chern insulators.
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Affiliation(s)
- F Nur Ünal
- TCM Group, Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
| | - Adrien Bouhon
- Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden
- Department of Physics and Astronomy, Uppsala University, Box 516, SE-751 21 Uppsala, Sweden
| | - Robert-Jan Slager
- TCM Group, Cavendish Laboratory, University of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom
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8
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Hu H, Zhao E. Topological Invariants for Quantum Quench Dynamics from Unitary Evolution. PHYSICAL REVIEW LETTERS 2020; 124:160402. [PMID: 32383903 DOI: 10.1103/physrevlett.124.160402] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/06/2019] [Accepted: 03/30/2020] [Indexed: 05/22/2023]
Abstract
Recent experiments began to explore the topological properties of quench dynamics, i.e., the time evolution following a sudden change in the Hamiltonian, via tomography of quantum gases in optical lattices. In contrast to the well-established theory for static band insulators or periodically driven systems, at present it is not clear whether, and how, topological invariants can be defined for a general quench of band insulators. Previous work solved a special case of this problem beautifully using Hopf mapping of two-band Hamiltonians in two dimensions. However, it only works for a topologically trivial initial state and is hard to generalize to multiband systems or other dimensions. Here we introduce the concept of loop unitary constructed from the unitary time-evolution operator and show its homotopy invariant fully characterizes the dynamical topology. For two-band systems in two dimensions, we prove that the invariant is precisely equal to the change in the Chern number across the quench, regardless of the initial state. We further show that the nontrivial dynamical topology manifests as hedgehog defects in the loop unitary and also as winding and linking of its eigenvectors along a curve where dynamical quantum phase transition occurs. This opens up a systematic route to classify and characterize quantum quench dynamics.
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Affiliation(s)
- Haiping Hu
- Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USA
- Department of Physics and Astronomy, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA
| | - Erhai Zhao
- Department of Physics and Astronomy, George Mason University, Fairfax, Virginia 22030, USA
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9
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Xie D, Deng TS, Xiao T, Gou W, Chen T, Yi W, Yan B. Topological Quantum Walks in Momentum Space with a Bose-Einstein Condensate. PHYSICAL REVIEW LETTERS 2020; 124:050502. [PMID: 32083915 DOI: 10.1103/physrevlett.124.050502] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/07/2019] [Accepted: 01/15/2020] [Indexed: 05/22/2023]
Abstract
We report the experimental implementation of discrete-time topological quantum walks of a Bose-Einstein condensate in momentum space. Introducing stroboscopic driving sequences to the generation of a momentum lattice, we show that the dynamics of atoms along the lattice is effectively governed by a periodically driven Su-Schrieffer-Heeger model, which is equivalent to a discrete-time topological quantum walk. We directly measure the underlying topological invariants through time-averaged mean chiral displacements, which are consistent with our experimental observation of topological phase transitions. We then observe interaction-induced localization in the quantum-walk dynamics, where atoms tend to populate a single momentum-lattice site under interactions that are nonlocal in momentum space. Our experiment opens up the avenue of investigating discrete-time topological quantum walks using cold atoms, where the many-body environment and tunable interactions offer exciting new possibilities.
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Affiliation(s)
- Dizhou Xie
- Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device of Physics Department, Zhejiang University, Hangzhou 310027, China
| | - Tian-Shu Deng
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
| | - Teng Xiao
- Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device of Physics Department, Zhejiang University, Hangzhou 310027, China
| | - Wei Gou
- Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device of Physics Department, Zhejiang University, Hangzhou 310027, China
| | - Tao Chen
- Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device of Physics Department, Zhejiang University, Hangzhou 310027, China
| | - Wei Yi
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei 230026, China
| | - Bo Yan
- Interdisciplinary Center of Quantum Information, State Key Laboratory of Modern Optical Instrumentation, and Zhejiang Province Key Laboratory of Quantum Technology and Device of Physics Department, Zhejiang University, Hangzhou 310027, China
- Collaborative Innovation Centre of Advanced Microstructures, Nanjing University, Nanjing 210093, China
- Key Laboratory of Quantum Optics, Chinese Academy of Sciences, Shanghai 200800, China
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10
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Barbarino S, Yu J, Zoller P, Budich JC. Preparing Atomic Topological Quantum Matter by Adiabatic Nonunitary Dynamics. PHYSICAL REVIEW LETTERS 2020; 124:010401. [PMID: 31976708 DOI: 10.1103/physrevlett.124.010401] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2019] [Indexed: 06/10/2023]
Abstract
Motivated by the outstanding challenge of realizing low-temperature states of quantum matter in synthetic materials, we propose and study an experimentally feasible protocol for preparing topological states such as Chern insulators. By definition, such (nonsymmetry protected) topological phases cannot be attained without going through a phase transition in a closed system, largely preventing their preparation in coherent dynamics. To overcome this fundamental caveat, we propose to couple the target system to a conjugate system, so as to prepare a symmetry protected topological phase in an extended system by intermittently breaking the protecting symmetry. Finally, the decoupled conjugate system is discarded, thus projecting onto the desired topological state in the target system. By construction, this protocol may be immediately generalized to the class of invertible topological phases, characterized by the existence of an inverse topological order. We illustrate our findings with microscopic simulations on an experimentally realistic Chern insulator model of ultracold fermionic atoms in a driven spin-dependent hexagonal optical lattice.
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Affiliation(s)
- S Barbarino
- Institute of Theoretical Physics, Technische Universität Dresden and Würzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany
| | - J Yu
- Center for Quantum Physics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - P Zoller
- Center for Quantum Physics, Faculty of Mathematics, Computer Science and Physics, University of Innsbruck, Innsbruck A-6020, Austria
- Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria
| | - J C Budich
- Institute of Theoretical Physics, Technische Universität Dresden and Würzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany
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11
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Qiu X, Deng TS, Hu Y, Xue P, Yi W. Fixed Points and Dynamic Topological Phenomena in a Parity-Time-Symmetric Quantum Quench. iScience 2019; 20:392-401. [PMID: 31622880 PMCID: PMC6818370 DOI: 10.1016/j.isci.2019.09.037] [Citation(s) in RCA: 23] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/01/2019] [Revised: 08/23/2019] [Accepted: 09/24/2019] [Indexed: 11/26/2022] Open
Abstract
We identify dynamic topological phenomena such as dynamic Chern numbers and dynamic quantum phase transitions in quantum quenches of the non-Hermitian Su-Schrieffer-Heeger Hamiltonian with parity-time (PT) symmetry. Their occurrences in the non-unitary dynamics are intimately connected with fixed points in the Brillouin zone, where the density matrices do not evolve in time. Based on our theoretical formalism characterizing topological properties of non-unitary dynamics, we prove the existence of fixed points for quenches between distinct static topological phases in the PT-symmetry-preserving regime, thus unveiling the interplay between dynamic topological phenomena and PT symmetry. Interestingly, non-Hermiticity of the driving Hamiltonian gives rise to rich dynamic topological phenomena which are different, either qualitatively or quantitatively, from their counterparts in unitary dynamics. Our work sheds light on dynamic topological phenomena in open systems and is readily accessible in experiments.
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Affiliation(s)
- Xingze Qiu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
| | - Tian-Shu Deng
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China
| | - Ying Hu
- State Key Laboratory of Quantum Optics and Quantum Optics Devices, Institute of Laser Spectroscopy, Shanxi University, Taiyuan, Shanxi 030006, China; Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan, Shanxi 030006, China.
| | - Peng Xue
- Beijing Computational Science Research Center, Beijing 100084, China; Department of Physics, Southeast University, Nanjing 211189, China; State Key Laboratory of Precision Spectroscopy, East China Normal University, Shanghai 200062, China.
| | - Wei Yi
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China; CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei 230026, China.
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12
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Wang K, Qiu X, Xiao L, Zhan X, Bian Z, Sanders BC, Yi W, Xue P. Observation of emergent momentum-time skyrmions in parity-time-symmetric non-unitary quench dynamics. Nat Commun 2019; 10:2293. [PMID: 31123259 PMCID: PMC6533298 DOI: 10.1038/s41467-019-10252-7] [Citation(s) in RCA: 67] [Impact Index Per Article: 13.4] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2018] [Accepted: 04/30/2019] [Indexed: 11/09/2022] Open
Abstract
Topology in quench dynamics gives rise to intriguing dynamic topological phenomena, which are intimately connected to the topology of static Hamiltonians yet challenging to probe experimentally. Here we theoretically characterize and experimentally detect momentum-time skyrmions in parity-time [Formula: see text]-symmetric non-unitary quench dynamics in single-photon discrete-time quantum walks. The emergent skyrmion structures are protected by dynamic Chern numbers defined for the emergent two-dimensional momentum-time submanifolds, and are revealed through our experimental scheme enabling the construction of time-dependent non-Hermitian density matrices via direct measurements in position space. Our work experimentally reveals the interplay of [Formula: see text] symmetry and quench dynamics in inducing emergent topological structures, and highlights the application of discrete-time quantum walks for the study of dynamic topological phenomena.
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Affiliation(s)
- Kunkun Wang
- Beijing Computational Science Research Center, 100084, Beijing, China
- Department of Physics, Southeast University, 211189, Nanjing, China
| | - Xingze Qiu
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China
- CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China
| | - Lei Xiao
- Beijing Computational Science Research Center, 100084, Beijing, China
- Department of Physics, Southeast University, 211189, Nanjing, China
| | - Xiang Zhan
- Beijing Computational Science Research Center, 100084, Beijing, China
- Department of Physics, Southeast University, 211189, Nanjing, China
| | - Zhihao Bian
- Beijing Computational Science Research Center, 100084, Beijing, China
- Department of Physics, Southeast University, 211189, Nanjing, China
| | - Barry C Sanders
- Institute for Quantum Science and Technology, University of Calgary, Alberta, T2N 1N4, Canada
- Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, 201315, Shanghai, China
| | - Wei Yi
- CAS Key Laboratory of Quantum Information, University of Science and Technology of China, 230026, Hefei, China.
- CAS Center For Excellence in Quantum Information and Quantum Physics, Hefei, 230026, China.
| | - Peng Xue
- Beijing Computational Science Research Center, 100084, Beijing, China.
- Department of Physics, Southeast University, 211189, Nanjing, China.
- State Key Laboratory of Precision Spectroscopy, East China Normal University, 200062, Shanghai, China.
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13
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Tarnowski M, Ünal FN, Fläschner N, Rem BS, Eckardt A, Sengstock K, Weitenberg C. Measuring topology from dynamics by obtaining the Chern number from a linking number. Nat Commun 2019; 10:1728. [PMID: 30988292 PMCID: PMC6465319 DOI: 10.1038/s41467-019-09668-y] [Citation(s) in RCA: 36] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/29/2018] [Accepted: 03/20/2019] [Indexed: 11/18/2022] Open
Abstract
Integer-valued topological indices, characterizing nonlocal properties of quantum states of matter, are known to directly predict robust physical properties of equilibrium systems. The Chern number, e.g., determines the quantized Hall conductivity of an insulator. Using non-interacting fermionic atoms in a periodically driven optical lattice, here we demonstrate experimentally that the Chern number determines also the far-from-equilibrium dynamics of a quantum system. Extending a respective proposal to Floquet systems, we measure the linking number that characterizes the trajectories of momentum-space vortices emerging after a strong quench. We observe that it directly corresponds to the ground-state Chern number. This one-to-one relation between a dynamical and a static topological index allows us to experimentally map out the phase diagram of our system. Furthermore, we measure the instantaneous Chern number and show that it remains zero under the unitary dynamics. The connection between the topological properties of the ground state and non-equilibrium dynamics remains obscure. Here, Tarnowski et al. define and measure a linking number between static and dynamical vortices, which directly corresponds to the ground-state Chern number.
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Affiliation(s)
- Matthias Tarnowski
- Institut für Laserphysik, Universität Hamburg, 22761, Hamburg, Germany.,The Hamburg Centre for Ultrafast Imaging, 22761, Hamburg, Germany
| | - F Nur Ünal
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187, Dresden, Germany
| | - Nick Fläschner
- Institut für Laserphysik, Universität Hamburg, 22761, Hamburg, Germany.,The Hamburg Centre for Ultrafast Imaging, 22761, Hamburg, Germany
| | - Benno S Rem
- Institut für Laserphysik, Universität Hamburg, 22761, Hamburg, Germany.,The Hamburg Centre for Ultrafast Imaging, 22761, Hamburg, Germany
| | - André Eckardt
- Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187, Dresden, Germany
| | - Klaus Sengstock
- Institut für Laserphysik, Universität Hamburg, 22761, Hamburg, Germany. .,The Hamburg Centre for Ultrafast Imaging, 22761, Hamburg, Germany. .,Zentrum für Optische Quantentechnologien, Universität Hamburg, 22761, Hamburg, Germany.
| | - Christof Weitenberg
- Institut für Laserphysik, Universität Hamburg, 22761, Hamburg, Germany.,The Hamburg Centre for Ultrafast Imaging, 22761, Hamburg, Germany
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14
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Chu SK, Zhu G, Garrison JR, Eldredge Z, Curiel AV, Bienias P, Spielman IB, Gorshkov AV. Scale-Invariant Continuous Entanglement Renormalization of a Chern Insulator. PHYSICAL REVIEW LETTERS 2019; 122:120502. [PMID: 30978046 PMCID: PMC6990462 DOI: 10.1103/physrevlett.122.120502] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/14/2018] [Indexed: 06/09/2023]
Abstract
The multiscale entanglement renormalization ansatz (MERA) postulates the existence of quantum circuits that renormalize entanglement in real space at different length scales. Chern insulators, however, cannot have scale-invariant discrete MERA circuits with a finite bond dimension. In this Letter, we show that the continuous MERA (cMERA), a modified version of MERA adapted for field theories, possesses a fixed point wave function with a nonzero Chern number. Additionally, it is well known that reversed MERA circuits can be used to prepare quantum states efficiently in time that scales logarithmically with the size of the system. However, state preparation via MERA typically requires the advent of a full-fledged universal quantum computer. In this Letter, we demonstrate that our cMERA circuit can potentially be realized in existing analog quantum computers, i.e., an ultracold atomic Fermi gas in an optical lattice with light-induced spin-orbit coupling.
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Affiliation(s)
- Su-Kuan Chu
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Guanyu Zhu
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - James R Garrison
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Zachary Eldredge
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Ana Valdés Curiel
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Przemyslaw Bienias
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - I B Spielman
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
| | - Alexey V Gorshkov
- Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA
- Joint Center for Quantum Information and Computer Science, NIST/University of Maryland, College Park, Maryland 20742, USA
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15
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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: 123] [Impact Index Per Article: 24.6] [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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16
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Sun W, Yi CR, Wang BZ, Zhang WW, Sanders BC, Xu XT, Wang ZY, Schmiedmayer J, Deng Y, Liu XJ, Chen S, Pan JW. Uncover Topology by Quantum Quench Dynamics. PHYSICAL REVIEW LETTERS 2018; 121:250403. [PMID: 30608809 DOI: 10.1103/physrevlett.121.250403] [Citation(s) in RCA: 34] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/10/2018] [Revised: 08/21/2018] [Indexed: 05/22/2023]
Abstract
Topological quantum states are characterized by nonlocal invariants. We present a new dynamical approach for ultracold-atom systems to uncover their band topology, and we provide solid evidence to demonstrate its experimental advantages. After quenching a two-dimensional (2D) Chern band, realized in an ultracold ^{87}Rb gas from a trivial to a topological parameter regime, we observe an emerging ring structure in the spin dynamics during the unitary evolution, which uniquely corresponds to the Chern number for the postquench band. By extracting 2D bulk topology from the 1D ring pattern, our scheme displays simplicity and is insensitive to perturbations. This insensitivity enables a high-precision determination of the full phase diagram for the system's band topology.
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Affiliation(s)
- Wei Sun
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
| | - Chang-Rui Yi
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
| | - Bao-Zong Wang
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Wei-Wei Zhang
- Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, New South Wales 2006, Australia
| | - Barry C Sanders
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Institute for Quantum Science and Technology, University of Calgary, Calgary, Alberta T2N 1N4, Canada
- Program in Quantum Information Science, Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
| | - Xiao-Tian Xu
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
| | - Zong-Yao Wang
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
| | - Joerg Schmiedmayer
- Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, Stadionallee 2, 1020 Vienna, Austria
| | - Youjin Deng
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
| | - Xiong-Jun Liu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Shuai Chen
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
| | - Jian-Wei Pan
- Shanghai Branch, National Laboratory for Physical Sciences at Microscale and Department of Modern Physics, University of Science and Technology of China, Shanghai 201315, China
- Chinese Academy of Sciences Center for Excellence: Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei Anhui 230326, China
- CAS-Alibaba Lab for Quantum Computation, Shanghai 201315, China
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17
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Gong Z, Ueda M. Topological Entanglement-Spectrum Crossing in Quench Dynamics. PHYSICAL REVIEW LETTERS 2018; 121:250601. [PMID: 30608813 DOI: 10.1103/physrevlett.121.250601] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/17/2017] [Revised: 03/29/2018] [Indexed: 05/22/2023]
Abstract
We unveil the stable (d+1)-dimensional topological structures underlying the quench dynamics for all of the Altland-Zirnbauer classes in d=1 dimension, and we propose to detect such dynamical topology from the time evolution of entanglement spectra. Focusing on systems in classes BDI and D, we find crossings in single-particle entanglement spectra for quantum quenches between different symmetry-protected topological phases. The entanglement-spectrum crossings are shown to be stable against symmetry-preserving disorder and faithfully reflect both Z (class BDI) and Z_{2} (class D) topological characterizations. As a by-product, we unravel the topological origin of the global degeneracies temporarily emerging in the many-body entanglement spectrum in the quench dynamics of the transverse-field Ising model. These findings can experimentally be tested in ultracold atoms and trapped ions with the help of cutting-edge tomography for quantum many-body states. Our work paves the way towards a systematic understanding of the role of topology in quench dynamics.
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Affiliation(s)
- Zongping Gong
- 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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18
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Zhang L, Zhang L, Niu S, Liu XJ. Dynamical classification of topological quantum phases. Sci Bull (Beijing) 2018; 63:1385-1391. [PMID: 36658977 DOI: 10.1016/j.scib.2018.09.018] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/19/2018] [Revised: 09/21/2018] [Accepted: 09/21/2018] [Indexed: 01/21/2023]
Abstract
Topological phase of matter is now a mainstream of research in condensed matter physics, of which the classification, synthesis, and detection of topological states have brought excitements over the recent decade while remain incomplete with ongoing challenges in both theory and experiment. Here we propose to establish a universal non-equilibrium characterization of the equilibrium topological quantum phases classified by integers, and further propose the high-precision dynamical schemes to detect such states. The framework of the dynamical classification theory consists of basic theorems. First, we uncover that classifying a d-dimensional (dD) gapped topological phase of generic multibands can reduce to a (d-1)D invariant defined on so-called band inversion surfaces (BISs), rendering a bulk-surface duality which simplifies the topological characterization. Further, we show in quenching across phase boundary the (pseudo) spin dynamics to exhibit unique topological patterns on BISs, which are attributed to the post-quench bulk topology and manifest a dynamical bulk-surface correspondence. For this the topological phase is classified by a dynamical topological invariant measured from an emergent dynamical spin-texture field on the BISs. Applications to quenching experiments on feasible models are proposed and studied, demonstrating the new experimental strategies to detect topological phases with high feasibility. This work opens a broad new direction to classify and detect topological phases by non-equilibrium quantum dynamics.
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Affiliation(s)
- Lin Zhang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China; Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Long Zhang
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China; Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Sen Niu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China; Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
| | - Xiong-Jun Liu
- International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China; Collaborative Innovation Center of Quantum Matter, Beijing 100871, China; Institute for Quantum Science and Engineering and Department of Physics, Southern University of Science and Technology, Shenzhen 518055, China.
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19
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McGinley M, Cooper NR. Topology of One-Dimensional Quantum Systems Out of Equilibrium. PHYSICAL REVIEW LETTERS 2018; 121:090401. [PMID: 30230907 DOI: 10.1103/physrevlett.121.090401] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/16/2018] [Indexed: 06/08/2023]
Abstract
We study the topological properties of one-dimensional systems undergoing unitary time evolution. We show that symmetries possessed both by the initial wave function and by the Hamiltonian at all times may not be present in the time-dependent wave function-a phenomenon which we dub "dynamically induced symmetry breaking." This leads to the possibility of a time-varying bulk index after quenching within noninteracting gapped topological phases. The consequences are observable experimentally through particle transport measurements. With reference to the entanglement spectrum, we explain how the topology of the wave function can change out of equilibrium, both for noninteracting fermions and for symmetry-protected topological phases protected by antiunitary symmetries.
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Affiliation(s)
- Max McGinley
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
| | - Nigel R Cooper
- T.C.M. Group, Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge, CB3 0HE, United Kingdom
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20
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Bandyopadhyay S, Laha S, Bhattacharya U, Dutta A. Exploring the possibilities of dynamical quantum phase transitions in the presence of a Markovian bath. Sci Rep 2018; 8:11921. [PMID: 30093653 PMCID: PMC6085341 DOI: 10.1038/s41598-018-30377-x] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2018] [Accepted: 07/19/2018] [Indexed: 11/10/2022] Open
Abstract
We explore the possibility of dynamical quantum phase transitions (DQPTs) occurring during the temporal evolution of a quenched transverse field Ising chain coupled to a particle loss type of bath (local in Jordan-Wigner fermion space) using two versions of the Loschmidt overlap (LO), namely, the fidelity induced LO and the interferometric phase induced LO. The bath, on the one hand, dictates the dissipative evolution following a sudden quench and on the other, plays a role in dissipative mixed state preparation in the later part of the study. During a dissipative evolution following a sudden quench, no trace of DQPTs are revealed in both the fidelity and the interferometric phase approaches; however, remarkably the interferometric phase approach reveals the possibility of inter-steady state DQPTs in passage from one steady state to the other when the system is subjected to a quench after having reached the first steady state. We further probe the occurrences of DQPTs when the system evolves unitarily after being prepared in a mixed state of engineered purity by ramping the transverse field in a linear fashion in the presence of the bath. In this case though the fidelity approach fails to indicate any DQPT, the interferometric approach indeed unravels the possibility of occurrence of DQPTs which persists even up to a considerable loss of purity of the engineered initial state as long as a constraint relation involving the dissipative coupling and ramping time (rate) is satisfied. This constraint relation also marks the boundary between two dynamically inequivalent phases; in one the LO vanishes for the critical momentum mode (and hence DQPTs exist) while in the other no such critical mode can exist and hence the LO never vanishes.
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Affiliation(s)
| | - Sudarshana Laha
- Department of Physics, Indian Institute of Technology, Kanpur, 208016, India
| | - Utso Bhattacharya
- Department of Physics, Indian Institute of Technology, Kanpur, 208016, India
| | - Amit Dutta
- Department of Physics, Indian Institute of Technology, Kanpur, 208016, India
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21
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Galilo B, Lee DKK, Barnett R. Topological Edge-State Manifestation of Interacting 2D Condensed Boson-Lattice Systems in a Harmonic Trap. PHYSICAL REVIEW LETTERS 2017; 119:203204. [PMID: 29219366 DOI: 10.1103/physrevlett.119.203204] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/09/2017] [Indexed: 06/07/2023]
Abstract
In this Letter, it is shown that interactions can facilitate the emergence of topological edge states of quantum-degenerate bosonic systems in the presence of a harmonic potential. This effect is demonstrated with the concrete model of a hexagonal lattice populated by spin-one bosons under a synthetic gauge field. In fermionic or noninteracting systems, the presence of a harmonic trap can obscure the observation of edge states. For our system with weakly interacting bosons in the Thomas-Fermi regime, we can clearly see a topological band structure with a band gap traversed by edge states. We also find that the number of edge states crossing the gap is increased in the presence of a harmonic trap, and the edge modes experience an energy shift while traversing the first Brillouin zone which is related to the topological properties of the system. We find an analytical expression for the edge-state energies and our comparison with numerical computation shows excellent agreement.
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Affiliation(s)
- Bogdan Galilo
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
| | - Derek K K Lee
- Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom
| | - Ryan Barnett
- Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
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22
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Wang C, Zhang P, Chen X, Yu J, Zhai H. Scheme to Measure the Topological Number of a Chern Insulator from Quench Dynamics. PHYSICAL REVIEW LETTERS 2017; 118:185701. [PMID: 28524691 DOI: 10.1103/physrevlett.118.185701] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/30/2016] [Indexed: 05/22/2023]
Abstract
We show how the topological number of a static Hamiltonian can be measured from a dynamical quench process. We focus on a two-band Chern insulator in two dimension, for instance, the Haldane model, whose dynamical process can be described by a mapping from the [k_{x},k_{y},t] space to the Bloch sphere, characterized by the Hopf invariant. Such a mapping has been constructed experimentally by measurements in cold atom systems. We show that, taking any two constant vectors on the Bloch sphere, their inverse images of this mapping are two trajectories in the [k_{x},k_{y},t] space, and the linking number of these two trajectories exactly equals the Chern number of the static Hamiltonian. Applying this result to a recent experiment from the Hamburg group, we show that the linking number of the trajectories of the phase vortices determines the phase boundary of the static Hamiltonian.
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Affiliation(s)
- Ce Wang
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Pengfei Zhang
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Xin Chen
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Jinlong Yu
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
| | - Hui Zhai
- Institute for Advanced Study, Tsinghua University, Beijing 100084, China
- Collaborative Innovation Center of Quantum Matter, Beijing 100084, China
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23
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Wilson JH, Song JCW, Refael G. Remnant Geometric Hall Response in a Quantum Quench. PHYSICAL REVIEW LETTERS 2016; 117:235302. [PMID: 27982622 DOI: 10.1103/physrevlett.117.235302] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/29/2016] [Indexed: 05/22/2023]
Abstract
Out-of-equilibrium systems can host phenomena that transcend the usual restrictions of equilibrium systems. Here, we unveil how out-of-equilibrium states, prepared via a quantum quench in a two-band system, can exhibit a nonzero Hall-type current-a remnant Hall response-even when the instantaneous Hamiltonian is time reversal symmetric (in contrast to equilibrium Hall currents). Interestingly, the remnant Hall response arises from the coherent dynamics of the wave function that retain a remnant of its quantum geometry postquench, and can be traced to processes beyond linear response. Quenches in two-band Dirac systems are natural venues for realizing remnant Hall currents, which exist when either mirror or time-reversal symmetry are broken (before or after the quench). Its long time persistence, sensitivity to symmetry breaking, and decoherence-type relaxation processes allow it to be used as a sensitive diagnostic of the complex out-of-equilibrium dynamics readily controlled and probed in cold-atomic optical lattice experiments.
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Affiliation(s)
- Justin H Wilson
- Institute of Quantum Information and Matter and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
| | - Justin C W Song
- Institute of Quantum Information and Matter and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
- Walter Burke Institute of Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
| | - Gil Refael
- Institute of Quantum Information and Matter and Department of Physics, California Institute of Technology, Pasadena, California 91125, USA
- Walter Burke Institute of Theoretical Physics, California Institute of Technology, Pasadena, California 91125, USA
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24
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Hu Y, Zoller P, Budich JC. Dynamical Buildup of a Quantized Hall Response from Nontopological States. PHYSICAL REVIEW LETTERS 2016; 117:126803. [PMID: 27689290 DOI: 10.1103/physrevlett.117.126803] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2016] [Indexed: 05/22/2023]
Abstract
We consider a two-dimensional system initialized in a topologically trivial state before its Hamiltonian is ramped through a phase transition into a Chern insulator regime. This scenario is motivated by current experiments with ultracold atomic gases aimed at realizing time-dependent dynamics in topological insulators. Our main findings are twofold. First, considering coherent dynamics, the nonequilibrium Hall response is found to approach a topologically quantized time-averaged value in the limit of slow but nonadiabatic parameter ramps, even though the Chern number of the state remains trivial. Second, adding dephasing, the destruction of quantum coherence is found to stabilize this Hall response, while the Chern number generically becomes undefined. We provide a geometric picture of this phenomenology in terms of the time-dependent Berry curvature.
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Affiliation(s)
- Ying Hu
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
| | - Peter Zoller
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
- Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
| | - Jan Carl Budich
- Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, 6020 Innsbruck, Austria
- Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
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