Topological classification of quasiperiodically driven quantum systems
Abstract
Few-level quantum systems driven by nf incommensurate fundamental frequencies exhibit temporal analogs of noninteracting phenomena in nf spatial dimensions, a consequence of the generalization of Floquet theory in frequency space. In this work, we organize the fundamental solutions of the frequency lattice model for nf = 2 into a quasienergy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasiperiodic dynamics is qualitatively similar to Floquet dynamics. The topological class with nonzero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasienergy spectrum. Nevertheless, using the case study of a spin in a quasiperiodically varying magnetic field, we show that topological class can be realized at low frequencies as a prethermal phase, and at finite frequencies using counterdiabatic tools.
- Authors:
-
- Boston Univ., MA (United States)
- Argonne National Lab. (ANL), Argonne, IL (United States)
- Publication Date:
- Research Org.:
- Argonne National Laboratory (ANL), Argonne, IL (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22). Materials Sciences & Engineering Division
- OSTI Identifier:
- 1500004
- Grant/Contract Number:
- AC02-06CH11357
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review B
- Additional Journal Information:
- Journal Volume: 99; Journal Issue: 6; Journal ID: ISSN 2469-9950
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Crowley, P. J. D., Martin, I., and Chandran, A. Topological classification of quasiperiodically driven quantum systems. United States: N. p., 2019.
Web. doi:10.1103/PhysRevB.99.064306.
Crowley, P. J. D., Martin, I., & Chandran, A. Topological classification of quasiperiodically driven quantum systems. United States. https://doi.org/10.1103/PhysRevB.99.064306
Crowley, P. J. D., Martin, I., and Chandran, A. Thu .
"Topological classification of quasiperiodically driven quantum systems". United States. https://doi.org/10.1103/PhysRevB.99.064306. https://www.osti.gov/servlets/purl/1500004.
@article{osti_1500004,
title = {Topological classification of quasiperiodically driven quantum systems},
author = {Crowley, P. J. D. and Martin, I. and Chandran, A.},
abstractNote = {Few-level quantum systems driven by nf incommensurate fundamental frequencies exhibit temporal analogs of noninteracting phenomena in nf spatial dimensions, a consequence of the generalization of Floquet theory in frequency space. In this work, we organize the fundamental solutions of the frequency lattice model for nf = 2 into a quasienergy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasiperiodic dynamics is qualitatively similar to Floquet dynamics. The topological class with nonzero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasienergy spectrum. Nevertheless, using the case study of a spin in a quasiperiodically varying magnetic field, we show that topological class can be realized at low frequencies as a prethermal phase, and at finite frequencies using counterdiabatic tools.},
doi = {10.1103/PhysRevB.99.064306},
journal = {Physical Review B},
number = 6,
volume = 99,
place = {United States},
year = {Thu Feb 28 00:00:00 EST 2019},
month = {Thu Feb 28 00:00:00 EST 2019}
}
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Works referenced in this record:
Universal high-frequency behavior of periodically driven systems: from dynamical stabilization to Floquet engineering
journal, March 2015
- Bukov, Marin; D'Alessio, Luca; Polkovnikov, Anatoli
- Advances in Physics, Vol. 64, Issue 2
Floquet Engineering of Correlated Tunneling in the Bose-Hubbard Model with Ultracold Atoms
journal, May 2016
- Meinert, F.; Mark, M. J.; Lauber, K.
- Physical Review Letters, Vol. 116, Issue 20
Floquet engineering with quasienergy bands of periodically driven optical lattices
journal, November 2015
- Holthaus, Martin
- Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 49, Issue 1
Steady states of interacting Floquet insulators
journal, January 2019
- Seetharam, Karthik I.; Bardyn, Charles-Edouard; Lindner, Netanel H.
- Physical Review B, Vol. 99, Issue 1
Optical Control of Topological Quantum Transport in Semiconductors
journal, July 2007
- Yao, Wang; MacDonald, A. H.; Niu, Qian
- Physical Review Letters, Vol. 99, Issue 4
Photovoltaic Hall effect in graphene
journal, February 2009
- Oka, Takashi; Aoki, Hideo
- Physical Review B, Vol. 79, Issue 8
Photoinduced Transition between Conventional and Topological Insulators in Two-Dimensional Electronic Systems
journal, June 2010
- Inoue, Jun-ichi; Tanaka, Akihiro
- Physical Review Letters, Vol. 105, Issue 1
Topological characterization of periodically driven quantum systems
journal, December 2010
- Kitagawa, Takuya; Berg, Erez; Rudner, Mark
- Physical Review B, Vol. 82, Issue 23
Floquet Spectrum and Transport through an Irradiated Graphene Ribbon
journal, November 2011
- Gu, Zhenghao; Fertig, H. A.; Arovas, Daniel P.
- Physical Review Letters, Vol. 107, Issue 21
Floquet topological insulator in semiconductor quantum wells
journal, March 2011
- Lindner, Netanel H.; Refael, Gil; Galitski, Victor
- Nature Physics, Vol. 7, Issue 6
Transport properties of nonequilibrium systems under the application of light: Photoinduced quantum Hall insulators without Landau levels
journal, December 2011
- Kitagawa, Takuya; Oka, Takashi; Brataas, Arne
- Physical Review B, Vol. 84, Issue 23
Majorana Fermions in Equilibrium and in Driven Cold-Atom Quantum Wires
journal, June 2011
- Jiang, Liang; Kitagawa, Takuya; Alicea, Jason
- Physical Review Letters, Vol. 106, Issue 22
Optically Engineering the Topological Properties of a Spin Hall Insulator
journal, January 2012
- Dóra, Balázs; Cayssol, Jérôme; Simon, Ferenc
- Physical Review Letters, Vol. 108, Issue 5
Topological Floquet spectrum in three dimensions via a two-photon resonance
journal, June 2013
- Lindner, Netanel H.; Bergman, Doron L.; Refael, Gil
- Physical Review B, Vol. 87, Issue 23
Merging of Dirac points and Floquet topological transitions in ac-driven graphene
journal, December 2013
- Delplace, Pierre; Gómez-León, Álvaro; Platero, Gloria
- Physical Review B, Vol. 88, Issue 24
Modulated Floquet Topological Insulators
journal, January 2013
- Katan, Yaniv Tenenbaum; Podolsky, Daniel
- Physical Review Letters, Vol. 110, Issue 1
Materials Design from Nonequilibrium Steady States: Driven Graphene as a Tunable Semiconductor with Topological Properties
journal, April 2013
- Iadecola, Thomas; Campbell, David; Chamon, Claudio
- Physical Review Letters, Vol. 110, Issue 17
Anomalous Edge States and the Bulk-Edge Correspondence for Periodically Driven Two-Dimensional Systems
journal, July 2013
- Rudner, Mark S.; Lindner, Netanel H.; Berg, Erez
- Physical Review X, Vol. 3, Issue 3
Floquet topological insulators
journal, January 2013
- Cayssol, Jérôme; Dóra, Balázs; Simon, Ferenc
- physica status solidi (RRL) - Rapid Research Letters, Vol. 7, Issue 1-2
Counter-propagating Edge Modes and Topological Phases of a Kicked Quantum Hall System
journal, January 2014
- Lababidi, Mahmoud; Satija, Indubala I.; Zhao, Erhai
- Physical Review Letters, Vol. 112, Issue 2
Periodically Driven Quantum Systems: Effective Hamiltonians and Engineered Gauge Fields
journal, August 2014
- Goldman, N.; Dalibard, J.
- Physical Review X, Vol. 4, Issue 3
Floquet Fractional Chern Insulators
journal, April 2014
- Grushin, Adolfo G.; Gómez-León, Álvaro; Neupert, Titus
- Physical Review Letters, Vol. 112, Issue 15
Effective Theory of Floquet Topological Transitions
journal, December 2014
- Kundu, Arijit; Fertig, H. A.; Seradjeh, Babak
- Physical Review Letters, Vol. 113, Issue 23
Chiral symmetry and bulk-boundary correspondence in periodically driven one-dimensional systems
journal, September 2014
- Asbóth, J. K.; Tarasinski, B.; Delplace, P.
- Physical Review B, Vol. 90, Issue 12
Topological Index for Periodically Driven Time-Reversal Invariant 2D Systems
journal, March 2015
- Carpentier, David; Delplace, Pierre; Fruchart, Michel
- Physical Review Letters, Vol. 114, Issue 10
Effect of periodic kicking on Chern insulators
journal, March 2017
- Wang, L. C.; Li, X. P.; Li, C. F.
- Physical Review B, Vol. 95, Issue 10
Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time
journal, May 1965
- Shirley, Jon H.
- Physical Review, Vol. 138, Issue 4B
Steady States and Quasienergies of a Quantum-Mechanical System in an Oscillating Field
journal, June 1973
- Sambe, Hideo
- Physical Review A, Vol. 7, Issue 6
New Frontiers in Quantum Information With Atoms and Ions
journal, March 2004
- Cirac, J. Ignacio; Zoller, Peter
- Physics Today, Vol. 57, Issue 3
Observation of Coherent Oscillation of a Single Nuclear Spin and Realization of a Two-Qubit Conditional Quantum Gate
journal, September 2004
- Jelezko, F.; Gaebel, T.; Popa, I.
- Physical Review Letters, Vol. 93, Issue 13
Long-Lived Qubit Memory Using Atomic Ions
journal, August 2005
- Langer, C.; Ozeri, R.; Jost, J.
- Physical Review Letters, Vol. 95, Issue 6
Fault-tolerant architecture for quantum computation using electrically controlled semiconductor spins
journal, December 2005
- Taylor, J. M.; Engel, H.-A.; Dür, W.
- Nature Physics, Vol. 1, Issue 3, p. 177-183
Spin qubits in graphene quantum dots
journal, February 2007
- Trauzettel, Björn; Bulaev, Denis V.; Loss, Daniel
- Nature Physics, Vol. 3, Issue 3
Theory of the neutral nitrogen-vacancy center in diamond and its application to the realization of a qubit
journal, June 2009
- Gali, Adam
- Physical Review B, Vol. 79, Issue 23
Quantum simulations with trapped ions
journal, April 2012
- Blatt, R.; Roos, C. F.
- Nature Physics, Vol. 8, Issue 4
High-Fidelity Preparation, Gates, Memory, and Readout of a Trapped-Ion Quantum Bit
journal, November 2014
- Harty, T. P.; Allcock, D. T. C.; Ballance, C. J.
- Physical Review Letters, Vol. 113, Issue 22
Quantum information processing with superconducting circuits: a review
journal, September 2017
- Wendin, G.
- Reports on Progress in Physics, Vol. 80, Issue 10
Single-qubit quantum memory exceeding ten-minute coherence time
journal, September 2017
- Wang, Ye; Um, Mark; Zhang, Junhua
- Nature Photonics, Vol. 11, Issue 10
Semiclassical many-mode floquet theory
journal, April 1983
- Ho, Tak-San; Chu, Shih-I; Tietz, James V.
- Chemical Physics Letters, Vol. 96, Issue 4
Quasi-Periodically Driven Quantum Systems
journal, October 2016
- Verdeny, Albert; Puig, Joaquim; Mintert, Florian
- Zeitschrift für Naturforschung A, Vol. 71, Issue 10
Topological Frequency Conversion in Strongly Driven Quantum Systems
journal, October 2017
- Martin, Ivar; Refael, Gil; Halperin, Bertrand
- Physical Review X, Vol. 7, Issue 4
Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
journal, August 2006
- Qi, Xiao-Liang; Wu, Yong-Shi; Zhang, Shou-Cheng
- Physical Review B, Vol. 74, Issue 8
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
journal, December 2006
- Bernevig, B. A.; Hughes, T. L.; Zhang, S.-C.
- Science, Vol. 314, Issue 5806, p. 1757-1761
Shortcuts to Adiabaticity by Counterdiabatic Driving
journal, September 2013
- del Campo, Adolfo
- Physical Review Letters, Vol. 111, Issue 10
Minimizing irreversible losses in quantum systems by local counterdiabatic driving
journal, May 2017
- Sels, Dries; Polkovnikov, Anatoli
- Proceedings of the National Academy of Sciences, Vol. 114, Issue 20
Time-quasiperiodic topological superconductors with Majorana multiplexing
journal, December 2018
- Peng, Yang; Refael, Gil
- Physical Review B, Vol. 98, Issue 22
Logarithmically Slow Relaxation in Quasiperiodically Driven Random Spin Chains
journal, February 2018
- Dumitrescu, Philipp T.; Vasseur, Romain; Potter, Andrew C.
- Physical Review Letters, Vol. 120, Issue 7
Anderson Transition in a One-Dimensional System with Three Incommensurate Frequencies
journal, January 1989
- Casati, Giulio; Guarneri, Italo; Shepelyansky, D. L.
- Physical Review Letters, Vol. 62, Issue 4
Anderson transition in a three-dimensional kicked rotor
journal, March 2009
- Wang, Jiao; García-García, Antonio M.
- Physical Review E, Vol. 79, Issue 3
On the response of a two-level quantum system to a class of time-dependent quasiperiodic perturbations
journal, November 1988
- Luck, J. M.; Orland, H.; Smilansky, U.
- Journal of Statistical Physics, Vol. 53, Issue 3-4
Spectral and stability aspects of quantum chaos
journal, July 1991
- Jauslin, H. R.; Lebowitz, J. L.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 1, Issue 1
Floquet spectrum for two-level systems in quasiperiodic time-dependent fields
journal, July 1992
- Blekher, P. M.; Jauslin, H. R.; Lebowitz, J. L.
- Journal of Statistical Physics, Vol. 68, Issue 1-2
On Formal Quasi-Periodic Solutions of the SchrÖDinger Equation for a Two-Level System with a Hamiltonian Depending Quasi–Periodically on time
journal, January 2000
- Barata, JoÃO C. A.
- Reviews in Mathematical Physics, Vol. 12, Issue 01
Pure Point Spectrum for Two-Level Systems in a Strong Quasi-Periodic Field
journal, June 2004
- Gentile, Guido
- Journal of Statistical Physics, Vol. 115, Issue 5/6
Quantized Hall Conductance in a Two-Dimensional Periodic Potential
journal, August 1982
- Thouless, D. J.; Kohmoto, M.; Nightingale, M. P.
- Physical Review Letters, Vol. 49, Issue 6
Topological Insulators and Topological Superconductors
book, January 2013
- Bernevig, B. Andrei
- Princeton University Press
Fractional Quantum Hall States at Zero Magnetic Field
journal, June 2011
- Neupert, Titus; Santos, Luiz; Chamon, Claudio
- Physical Review Letters, Vol. 106, Issue 23
Wave functions for fractional Chern insulators
journal, March 2012
- McGreevy, John; Swingle, Brian; Tran, Ky-Anh
- Physical Review B, Vol. 85, Issue 12
Non-Adiabatic Crossing of Energy Levels
journal, September 1932
- Zener, C.
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 137, Issue 833
Quantum Spin Hall Effect in Graphene
journal, November 2005
- Kane, C. L.; Mele, E. J.
- Physical Review Letters, Vol. 95, Issue 22
Berry’s phase for energy bands in solids
journal, June 1989
- Zak, J.
- Physical Review Letters, Vol. 62, Issue 23
Zak phase and the existence of edge states in graphene
journal, November 2011
- Delplace, P.; Ullmo, D.; Montambaux, G.
- Physical Review B, Vol. 84, Issue 19
Topological singularities and the general classification of Floquet–Bloch systems
journal, December 2015
- Nathan, Frederik; Rudner, Mark S.
- New Journal of Physics, Vol. 17, Issue 12
Abelian Floquet symmetry-protected topological phases in one dimension
journal, September 2016
- Roy, Rahul; Harper, Fenner
- Physical Review B, Vol. 94, Issue 12
Classification of topological phases in periodically driven interacting systems
journal, May 2016
- Else, Dominic V.; Nayak, Chetan
- Physical Review B, Vol. 93, Issue 20
Classification of Interacting Topological Floquet Phases in One Dimension
journal, October 2016
- Potter, Andrew C.; Morimoto, Takahiro; Vishwanath, Ashvin
- Physical Review X, Vol. 6, Issue 4
Periodic table for Floquet topological insulators
journal, October 2017
- Roy, Rahul; Harper, Fenner
- Physical Review B, Vol. 96, Issue 15
Topological characterization of periodically-driven quantum systems
text, January 2010
- Kitagawa, Takuya; Berg, Erez; Rudner, Mark
- arXiv
Topological Floquet Spectrum in Three Dimensions via a Two-Photon Resonance
text, January 2011
- Lindner, Netanel H.; Bergman, Doron L.; Refael, Gil
- arXiv
Modulated Floquet Topological Insulators
text, January 2012
- Katan, Yaniv Tenenbaum; Podolsky, Daniel
- arXiv
Anomalous edge states and the bulk-edge correspondence for periodically-driven two dimensional systems
text, January 2012
- Rudner, Mark S.; Lindner, Netanel H.; Berg, Erez
- arXiv
Counter-propagating edge modes and topological phases of a kicked quantum Hall system
text, January 2013
- Lababidi, Mahmoud; Satija, Indubala I.; Zhao, Erhai
- arXiv
High-fidelity preparation, gates, memory and readout of a trapped-ion quantum bit
text, January 2014
- Harty, T. P.; Allcock, D. T. C.; Ballance, C. J.
- arXiv
Effective Theory of Floquet Topological Transitions
text, January 2014
- Kundu, Arijit; Fertig, H. A.; Seradjeh, Babak
- arXiv
Topological index for periodically driven time-reversal invariant 2D systems
text, January 2014
- Carpentier, David; Delplace, Pierre; Fruchart, Michel
- arXiv
Floquet engineering of correlated tunneling in the Bose-Hubbard model with ultracold atoms
text, January 2016
- Meinert, Florian; Mark, Manfred J.; Lauber, Katharina
- arXiv
Classification of topological phases in periodically driven interacting systems
text, January 2016
- Else, Dominic V.; Nayak, Chetan
- arXiv
Quasi-periodically driven quantum systems
text, January 2016
- Verdeny, Albert; Puig, Joaquim; Mintert, Florian
- arXiv
Quantum information processing with superconducting circuits: a review
text, January 2016
- Wendin, G.
- arXiv
Topological frequency conversion in strongly driven quantum systems
text, January 2016
- Martin, Ivar; Refael, Gil; Halperin, Bertrand
- arXiv
Logarithmically Slow Relaxation in Quasi-Periodically Driven Random Spin Chains
text, January 2017
- Dumitrescu, Philipp T.; Vasseur, Romain; Potter, Andrew C.
- arXiv
Time-quasiperiodic topological superconductors with Majorana Multiplexing
text, January 2018
- Peng, Yang; Refael, Gil
- arXiv
Steady state of interacting Floquet insulators
text, January 2018
- Seetharam, Karthik I.; Bardyn, Charles-Edouard; Lindner, Netanel H.
- arXiv
Topological quantization of the spin Hall effect in two-dimensional paramagnetic semiconductors
text, January 2005
- Qi, Xiao-Liang; Wu, Yong-Shi; Zhang, Shou-Cheng
- arXiv
Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
text, January 2006
- Bernevig, B. Andrei; Hughes, Taylor L.; Zhang, Shou-Cheng
- arXiv
Observation of coherent oscillation of a single nuclear spin and realization of a two-qubit conditional quantum gate
text, January 2004
- Jelezko, F.; Gaebel, T.; Popa, I.
- arXiv
Long-lived qubit memory using atomic ions
text, January 2005
- Langer, C.; Ozeri, R.; Jost, J. D.
- arXiv
Works referencing / citing this record:
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- Zhao, Hongzheng; Mintert, Florian; Knolle, Johannes
- Physical Review B, Vol. 100, Issue 13
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text, January 2019
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- arXiv