Analog of Hamilton-Jacobi theory for the time-evolution operator
Abstract
In this paper we develop an analog of Hamilton-Jacobi theory for the time-evolution operator of a quantum many-particle system. The theory offers a useful approach to develop approximations to the time-evolution operator, and also provides a unified framework and starting point for many well-known approximations to the time-evolution operator. In the important special case of periodically driven systems at stroboscopic times, we find relatively simple equations for the coupling constants of the Floquet Hamiltonian, where a straightforward truncation of the couplings leads to a powerful class of approximations. Using our theory, we construct a flow chart that illustrates the connection between various common approximations, which also highlights some missing connections and associated approximation schemes. These missing connections turn out to imply an analytically accessible approximation that is the “inverse” of a rotating frame approximation and thus has a range of validity complementary to it. Here, we numerically test the various methods on the one-dimensional Ising model to confirm the ranges of validity that one would expect from the approximations used. The theory provides a map of the relations between the growing number of approximations for the time-evolution operator. We describe these relations in a table showing the limitations and advantagesmore »
- Authors:
-
- Univ. of Texas, Austin, TX (United States); Northeastern Univ., Boston, MA (United States)
- Univ. of Texas, Austin, TX (United States); Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Univ. of Texas, Austin, TX (United States)
- Univ. of Texas, Austin, TX (United States); Northeastern Univ., Boston, MA (United States); Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
- Publication Date:
- Research Org.:
- Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States)
- Sponsoring Org.:
- USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); USDOE Office of Science (SC), Basic Energy Sciences (BES); Gordon and Betty Moore Foundation (GBMF); National Science Foundation (NSF)
- OSTI Identifier:
- 1546524
- Alternate Identifier(s):
- OSTI ID: 1545446
- Grant/Contract Number:
- AC05-00OR22725; DMR-1729588; GBMF5305
- Resource Type:
- Accepted Manuscript
- Journal Name:
- Physical Review A
- Additional Journal Information:
- Journal Volume: 100; Journal Issue: 1; Journal ID: ISSN 2469-9926
- Publisher:
- American Physical Society (APS)
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND SUPERFLUIDITY
Citation Formats
Vogl, Michael, Laurell, Pontus, Barr, Aaron D., and Fiete, Gregory A. Analog of Hamilton-Jacobi theory for the time-evolution operator. United States: N. p., 2019.
Web. doi:10.1103/PhysRevA.100.012132.
Vogl, Michael, Laurell, Pontus, Barr, Aaron D., & Fiete, Gregory A. Analog of Hamilton-Jacobi theory for the time-evolution operator. United States. https://doi.org/10.1103/PhysRevA.100.012132
Vogl, Michael, Laurell, Pontus, Barr, Aaron D., and Fiete, Gregory A. Mon .
"Analog of Hamilton-Jacobi theory for the time-evolution operator". United States. https://doi.org/10.1103/PhysRevA.100.012132. https://www.osti.gov/servlets/purl/1546524.
@article{osti_1546524,
title = {Analog of Hamilton-Jacobi theory for the time-evolution operator},
author = {Vogl, Michael and Laurell, Pontus and Barr, Aaron D. and Fiete, Gregory A.},
abstractNote = {In this paper we develop an analog of Hamilton-Jacobi theory for the time-evolution operator of a quantum many-particle system. The theory offers a useful approach to develop approximations to the time-evolution operator, and also provides a unified framework and starting point for many well-known approximations to the time-evolution operator. In the important special case of periodically driven systems at stroboscopic times, we find relatively simple equations for the coupling constants of the Floquet Hamiltonian, where a straightforward truncation of the couplings leads to a powerful class of approximations. Using our theory, we construct a flow chart that illustrates the connection between various common approximations, which also highlights some missing connections and associated approximation schemes. These missing connections turn out to imply an analytically accessible approximation that is the “inverse” of a rotating frame approximation and thus has a range of validity complementary to it. Here, we numerically test the various methods on the one-dimensional Ising model to confirm the ranges of validity that one would expect from the approximations used. The theory provides a map of the relations between the growing number of approximations for the time-evolution operator. We describe these relations in a table showing the limitations and advantages of many common approximations, as well as the approximations introduced in this paper.},
doi = {10.1103/PhysRevA.100.012132},
journal = {Physical Review A},
number = 1,
volume = 100,
place = {United States},
year = {Mon Jul 29 00:00:00 EDT 2019},
month = {Mon Jul 29 00:00:00 EDT 2019}
}
Web of Science
Figures / Tables:
Works referenced in this record:
Colloquium : Nonequilibrium dynamics of closed interacting quantum systems
journal, August 2011
- Polkovnikov, Anatoli; Sengupta, Krishnendu; Silva, Alessandro
- Reviews of Modern Physics, Vol. 83, Issue 3
Colloquium: Atomic quantum gases in periodically driven optical lattices
journal, March 2017
- Eckardt, André
- Reviews of Modern Physics, Vol. 89, Issue 1
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
Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time
journal, May 1965
- Shirley, Jon H.
- Physical Review, Vol. 138, Issue 4B
Equilibration and order in quantum Floquet matter
journal, April 2017
- Moessner, R.; Sondhi, S. L.
- Nature Physics, Vol. 13, Issue 5
Periodic Thermodynamics of Isolated Quantum Systems
journal, April 2014
- Lazarides, Achilleas; Das, Arnab; Moessner, Roderich
- Physical Review Letters, Vol. 112, Issue 15
Floquet Dynamics of Boundary-Driven Systems at Criticality
journal, June 2017
- Berdanier, William; Kolodrubetz, Michael; Vasseur, Romain
- Physical Review Letters, Vol. 118, Issue 26
Phase Structure of Driven Quantum Systems
journal, June 2016
- Khemani, Vedika; Lazarides, Achilleas; Moessner, Roderich
- Physical Review Letters, Vol. 116, Issue 25
Periodically driven ergodic and many-body localized quantum systems
journal, February 2015
- Ponte, Pedro; Chandran, Anushya; Papić, Z.
- Annals of Physics, Vol. 353
Many-Body Localization in Periodically Driven Systems
journal, April 2015
- Ponte, Pedro; Papić, Z.; Huveneers, François
- Physical Review Letters, Vol. 114, Issue 14
Exponentially Slow Heating in Periodically Driven Many-Body Systems
journal, December 2015
- Abanin, Dmitry A.; De Roeck, Wojciech; Huveneers, François
- Physical Review Letters, Vol. 115, Issue 25
Effective Hamiltonians, prethermalization, and slow energy absorption in periodically driven many-body systems
journal, January 2017
- Abanin, Dmitry A.; De Roeck, Wojciech; Ho, Wen Wei
- Physical Review B, Vol. 95, Issue 1
Rigorous Bound on Energy Absorption and Generic Relaxation in Periodically Driven Quantum Systems
journal, March 2016
- Mori, Takashi; Kuwahara, Tomotaka; Saito, Keiji
- Physical Review Letters, Vol. 116, Issue 12
Prethermal Phases of Matter Protected by Time-Translation Symmetry
journal, March 2017
- Else, Dominic V.; Bauer, Bela; Nayak, Chetan
- Physical Review X, Vol. 7, Issue 1
Many-body physics with ultracold gases
journal, July 2008
- Bloch, Immanuel; Dalibard, Jean; Zwerger, Wilhelm
- Reviews of Modern Physics, Vol. 80, Issue 3
Colloquium : Artificial gauge potentials for neutral atoms
journal, November 2011
- Dalibard, Jean; Gerbier, Fabrice; Juzeliūnas, Gediminas
- Reviews of Modern Physics, Vol. 83, Issue 4
Towards properties on demand in quantum materials
journal, October 2017
- Basov, D. N.; Averitt, R. D.; Hsieh, D.
- Nature Materials, Vol. 16, Issue 11
Dynamics and Control in Complex Transition Metal Oxides
journal, July 2014
- Zhang, J.; Averitt, R. D.
- Annual Review of Materials Research, Vol. 44, Issue 1
Electrodynamics of correlated electron materials
journal, June 2011
- Basov, D. N.; Averitt, Richard D.; van der Marel, Dirk
- Reviews of Modern Physics, Vol. 83, Issue 2
Ultrafast optical spectroscopy of strongly correlated materials and high-temperature superconductors: a non-equilibrium approach
journal, March 2016
- Giannetti, Claudio; Capone, Massimo; Fausti, Daniele
- Advances in Physics, Vol. 65, Issue 2
Emergent ultrafast phenomena in correlated oxides and heterostructures
journal, January 2017
- Gandolfi, M.; Celardo, G. L.; Borgonovi, F.
- Physica Scripta, Vol. 92, Issue 3
Observation of Photon-Assisted Tunneling in Optical Lattices
journal, February 2008
- Sias, C.; Lignier, H.; Singh, Y. P.
- Physical Review Letters, Vol. 100, Issue 4
Coherent Control of Dressed Matter Waves
journal, March 2009
- Zenesini, Alessandro; Lignier, Hans; Ciampini, Donatella
- Physical Review Letters, Vol. 102, Issue 10
Superfluid-Insulator Transition in a Periodically Driven Optical Lattice
journal, December 2005
- Eckardt, André; Weiss, Christoph; Holthaus, Martin
- Physical Review Letters, Vol. 95, Issue 26
Experimental Realization of Strong Effective Magnetic Fields in an Optical Lattice
journal, December 2011
- Aidelsburger, M.; Atala, M.; Nascimbène, S.
- Physical Review Letters, Vol. 107, Issue 25
Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms
journal, December 2014
- Aidelsburger, M.; Lohse, M.; Schweizer, C.
- Nature Physics, Vol. 11, Issue 2
Light-Induced Superconductivity in a Stripe-Ordered Cuprate
journal, January 2011
- Fausti, D.; Tobey, R. I.; Dean, N.
- Science, Vol. 331, Issue 6014
Photo-induced superconductivity
journal, November 2017
- Cavalleri, Andrea
- Contemporary Physics, Vol. 59, Issue 1
Ultrafast Switching to a Stable Hidden Quantum State in an Electronic Crystal
journal, April 2014
- Stojchevska, L.; Vaskivskyi, I.; Mertelj, T.
- Science, Vol. 344, Issue 6180
Observation of a discrete time crystal
journal, March 2017
- Zhang, J.; Hess, P. W.; Kyprianidis, A.
- Nature, Vol. 543, Issue 7644
Observation of discrete time-crystalline order in a disordered dipolar many-body system
journal, March 2017
- Choi, Soonwon; Choi, Joonhee; Landig, Renate
- Nature, Vol. 543, Issue 7644
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
Photovoltaic Hall effect in graphene
journal, February 2009
- Oka, Takashi; Aoki, Hideo
- Physical Review B, Vol. 79, Issue 8
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
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
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
Irradiated graphene as a tunable Floquet topological insulator
journal, September 2014
- Usaj, Gonzalo; Perez-Piskunow, P. M.; Foa Torres, L. E. F.
- Physical Review B, Vol. 90, Issue 11
The Radiation Theories of Tomonaga, Schwinger, and Feynman
journal, February 1949
- Dyson, F. J.
- Physical Review, Vol. 75, Issue 3
The Matrix in Quantum Electrodynamics
journal, June 1949
- Dyson, F. J.
- Physical Review, Vol. 75, Issue 11
Divergence of Perturbation Theory in Quantum Electrodynamics
journal, February 1952
- Dyson, F. J.
- Physical Review, Vol. 85, Issue 4
The Magnus expansion and some of its applications
journal, January 2009
- Blanes, S.; Casas, F.; Oteo, J. A.
- Physics Reports, Vol. 470, Issue 5-6
On the convergence of the magnus expansion for spin systems in periodic magnetic fields
journal, September 1984
- Fel'dman, E. B.
- Physics Letters A, Vol. 104, Issue 9
On the exponential solution of differential equations for a linear operator
journal, November 1954
- Magnus, Wilhelm
- Communications on Pure and Applied Mathematics, Vol. 7, Issue 4
Exponential infinite-product representations of the time-displacement operator
journal, July 1989
- Klarsfeld, S.; Oteo, J. A.
- Journal of Physics A: Mathematical and General, Vol. 22, Issue 14
On the Magnus, Wilcox and Fer operator equations
journal, December 2004
- Fernández, F. M.
- European Journal of Physics, Vol. 26, Issue 1
Exponential Operators and Parameter Differentiation in Quantum Physics
journal, April 1967
- Wilcox, R. M.
- Journal of Mathematical Physics, Vol. 8, Issue 4
Flow Equation Approach to Periodically Driven Quantum Systems
journal, May 2019
- Vogl, Michael; Laurell, Pontus; Barr, Aaron D.
- Physical Review X, Vol. 9, Issue 2
Flow-equations for Hamiltonians
journal, January 1994
- Wegner, Franz
- Annalen der Physik, Vol. 506, Issue 2
Accurate Effective Hamiltonians via Unitary Flow in Floquet Space
journal, October 2013
- Verdeny, Albert; Mielke, Andreas; Mintert, Florian
- Physical Review Letters, Vol. 111, Issue 17
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
Non-equilibrium fractional quantum Hall state of light
journal, June 2013
- Hafezi, Mohammad; Lukin, Mikhail D.; Taylor, Jacob M.
- New Journal of Physics, Vol. 15, Issue 6
Quantum feedback in a weakly driven cavity QED system
journal, August 2004
- Reiner, J. E.; Smith, W. P.; Orozco, L. A.
- Physical Review A, Vol. 70, Issue 2
Weakly driven quantum coherent ratchets in cold-atom systems
journal, August 2010
- Heimsoth, M.; Creffield, C. E.; Sols, F.
- Physical Review A, Vol. 82, Issue 2
Floquet–Magnus theory and generic transient dynamics in periodically driven many-body quantum systems
journal, April 2016
- Kuwahara, Tomotaka; Mori, Takashi; Saito, Keiji
- Annals of Physics, Vol. 367
On the structure and convergence of the symmetric Zassenhaus formula
journal, August 2017
- Arnal, Ana; Casas, Fernando; Chiralt, Cristina
- Computer Physics Communications, Vol. 217
The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
journal, March 1999
- Boyd, John P.
- Acta Applicandae Mathematicae, Vol. 56, Issue 1, p. 1-98
Scaling approach for the time-dependent Kondo model
journal, February 2011
- Tomaras, C.; Kehrein, S.
- EPL (Europhysics Letters), Vol. 93, Issue 4
QuSpin: a Python package for dynamics and exact diagonalisation of quantum many body systems part I: spin chains
journal, January 2017
- Weinberg, Phillip; Bukov, Marin
- SciPost Physics, Vol. 2, Issue 1
Quantum localization bounds Trotter errors in digital quantum simulation
journal, April 2019
- Heyl, Markus; Hauke, Philipp; Zoller, Peter
- Science Advances, Vol. 5, Issue 4
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