The aim of the project MA14-002 'Adaptive Time-Splitting for Many-Body Quantum Propagation'
supported by the Vienna Science and Technology Fund (WWTF)
is to understand non-equilibrium quantum many-body systems, which is one of the grand challenges in current
physics. The ability to describe time-dependent extended systems on a quantitative level promises to open
up new technological developments such as laser control of chemical reactions, femtosecond lightfield
electronics, and material design for sustainable energy. Most experimental probes of quantum many-body
systems rely on so-called pump-probe schemes. The system is driven out of equilibrium and the ensuing
dynamics is studied. While for weak pump and probe fields the dynamics can be understood within linear
response theory where essentially equilibrium (or near ground state) properties enter, stronger pumping
requires a full time-dependent propagation beyond the perturbative limit. While the high-dimensional
many-particle Schrödinger equation is too complex to be solved for more than a few particles in three
dimensions, practically all approximations to it give rise to coupled nonlinear partial differential equations in
space and time in reduced dimensions. Still, propagation of these equations constitutes a major challenge.
In this project we aim for the development and implementation of a highly efficient and accurate method,
the adaptive time-splitting method, for the propagation of quantum many-body systems presently out of
reach. We will investigate the efficiency, error, and speed up of this method and apply it to both fermionic
and bosonic systems of current interest.
So far, the project yielded the following articles and preprints and
conference presentations. A collection of splitting methods
constructed in the course of the project or taken from the literature and assessed as to their
merits for computational purposes can be found at
http://www.asc.tuwien.ac.at/~winfried/splitting/.
F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and
J. Burgdörfer,
The time-dependent two-particle reduced density matrix method: LiH in strong laser fields,
J. Phys.: Conf. Ser. 635(2015), 112084.
W. Auzinger,
Th. Kassebacher, O. Koch,
M. Thalhammer,
Adaptive Splitting Methods for Nonlinear Schrödinger Equations in the Semiclassical Regime,
Numer. Algorithms 72(2016), pp. 1-35.
W. Auzinger,
H. Hofstätter,
D. Ketcheson,
and O. Koch,
Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations.
Part I: Construction of Optimized Schemes and Pairs of Schemes,
BIT
57(2017), pp. 55-74, DOI=dx.doi.org/10.1007/s10543-016-0626-9.
W. Auzinger,
W. Herfort, O. Koch, and
M. Thalhammer,
The BCH-Formula and Order Conditions for Splitting Methods,
in: Lie Groups, Differential Equations, and Geometry, G. Falcone (ed.),
UNIPA Springer Series, Vol. XIV(2017), pp. 71-84.
T. Sato, K. L. Ishikawa, I. Brezinova, F. Lackner, S. Nagele, and
J. Burgdörfer,
Time-dependent complete-active-space self-consistent-field method for atoms: Application to high-harmonic generation,
Phys. Rev. A 94, 023405 (2016).
W. Auzinger,
H. Hofstätter, and O. Koch,
Symbolic Manipulation of Flows of Nonlinear
Evolution Equations, with Application in the
Analysis of Split-Step Time Integrators, Springer LNCS 9890(2016), pp. 43-57.
F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and
J. Burgdörfer,
Time-dependent two-particle reduced density matrix theory: Application to high-harmonic generation,
J. Phys.: Conf. Ser. 875(2017), 012011.
F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and
J. Burgdörfer,
Time-dependent two-particle reduced density matrix theory: Application to high-harmonic generation,
J. Phys.: Conf. Ser. 875(2017), 022032.
W. Auzinger,
I. Brezinova, H. Hofstätter, O. Koch, and M. Quell,
Practical Splitting Methods for the Adaptive Integration of Nonlinear Evolution Equations.
Part II: Comparisons of Local Error Estimation and Step-Selection Strategies, submitted.
You can download an extended preprint of this paper as a
.pdf file.
X.-M. Tong, S. Borbely, I. Brezinova, F. Lackner, S. Nagele, J. Feist, L. Nagy, K. Tokesi, N. Toshima, and
J. Burgdörfer,
Energy straggling cross section for antiproton-atom collisions,
J. Phys.: Conf. Ser. 875(2017), 092026.
F. Lackner, I. Brezinova, T. Sato, K. L. Ishikawa, and
J. Burgdörfer,
High-harmonic spectra from time-dependent two-particle reduced density matrix theory,
Phys. Rev. A 95(2017), 033414.
S. Donsa, H. Hofstätter, O. Koch,
J. Burgdörfer, and I. Brezinova,
Long-time expansion of a Bose-Einstein condensate: Observability of Anderson localization,
Phys. Rev. A 96(2017), 043630.
V. Stooß, S. M. Cavaletto, S. Donsa, A. Blättermann, P. Birk,
C. H. Keitel, I. Brezinova,
J. Burgdörfer,
C. Ott, and T.Pfeifer
Real-time reconstruction of complex non-equilibrium quantum dynamics of
matter, submitted.
W. Auzinger and O. Koch,
An improved local error estimator for symmetric time-stepping schemes,
Appl. Math. Lett.
82(2018), pp.106-110.
W. Auzinger,
I. Brezinova, H. Hofstätter, and O. Koch,
Efficient adaptive exponential time integrators for nonlinear Schrödinger equations with nonlocal potential, in preparation.
O. Koch,
Convergence of exponential Lawson-multistep methods for the MCTDHF equations, submitted.
W. Auzinger,
H. Hofstätter, and O. Koch,
Symmetrized local error estimators for time-reversible one-step methods
in nonlinear evolution equations, submitted.
O. Koch,
Adaptive Full Discretization for Nonlinear Evolution Equations,
University of Bath,
U.K., April 2016.
O. Koch (invited talk),
Reduced Order of the Local Error of Splitting for
Parabolic Problems,
ICNAAM 2016, Rhodes, Greece, September 2016.
Joint with W. Auzinger, H. Hofstätter,
and M. Thalhammer.
I. Brezinova (invited talk),
A new theoretical approach for accurate multi-electron dynamics in strong fields: the time-dependent two-particle reduced density matrix method,
International Workshop on Atomic Physics, Max-Plank-Institute for Physics of Complex Systems, Dresden, Germany, November 2016.
Joint with F. Lackner, T. Sato, K. L. Ishikawa, and
J. Burgdörfer.
I. Brezinova (invited talk),
Time-dependent two-particle reduced density matrix theory: Application to high-harmonic generation,
12th European Conference on Atoms Molecules and Photons (ECAMP12), Frankfurt/Main, Germany, September 2016.
Joint with F. Lackner, T. Sato, K. L. Ishikawa, and
J. Burgdörfer.
H. Hofstätter,
Time propagation for Schrödinger-type equations with expensive nonlinearity,
Austrian Numerical Analysis Day 2017,
Salzburg, May 2017.
Joint with O. Koch.
H. Hofstätter,
Time propagation for Schr&oulm;dinger-type equations with expensive nonlinearity,
University of Innsbruck,
Innsbruck, July 2017.
Joint with O. Koch.
O. Koch (invited talk),
Adaptive exponential splitting and Lawson methods for Schrödinger equations,
SciCADE 2017, Bath, U.K., September 2017.
Joint with W. Auzinger, I. Brezinova, H. Hofstätter,
and M. Thalhammer.
O. Koch,
Adaptive Exponential Integrators for Schrödinger Equations,
University of Gdansk, Gdansk, November 2017.
S. Donsa,
Anderson localization of a one-dimensional Bose-Einstein condensate after long-time expansion,
Vienna young Scientists Symposium, Vienna, June 2016.
Joint with H. Hofstätter, O. Koch,
J. Burgdörfer,
and I. Brezinova.
I. Brezinova (invited talk),
Expansion of Bose-Einstein condensates in disorder: Anderson localization and wave chaos,
International Workshop on Quantum-Classical Transition in Many-Body Systems, Dresden, Germany, February 2017.
Joint with S. Donsa, H. Hofstätter, O. Koch, and
J. Burgdörfer.
S. Donsa,
Ultrafast buildup of a Fano resonance in the time domain,
6th International Conference on Attosecond Physics, Xian, China, July 2017.
Joint with A. Kaldun, A. Blättermann, V. Stooß,
H. Wei, R. Pazourek, S. Nagele, C. Ott, C.-D. Lin,
J. Burgdörfer, and T. Pfeifer.
I. Brezinova (invited talk),
A new theoretical approach for accurate multi-electron dynamics in strong fields:
the time-dependent two-particle reduced density matrix method,
14th International Conference on Multiphoton Processes (ICOMP 2017), Budapest, Hungary, September 2017.
Joint with F. Lackner, S. Donsa, T. Sato, K. L. Ishikawa, and
J. Burgdörfer.
F. Lackner,
The time-dependent two-particle reduced density matrix method";
Vortrag: International Workshop on 'New challenges in Reduced Density Matrix Functional Theory:
Symmetries, time-evolution and entanglement',
CECAM-HQ-EPFL, Lausanne, Switzerland; September 2017.
Joint with S. Donsa, T. Sato, K. L. Ishikawa,
J. Burgdörfer, and
I. Brezinova.
S. Donsa (invited talk),
The time-dependent two-particle reduced density matrix method,
Informal seminar of the Departement of Theoretical Condensed Matter at the University Autonoma Madrid,
Madrid, Spain, November 2017.
Joint with F. Lackner, T. Sato, K. L. Ishikawa,
J. Burgdörfer, and
I. Brezinova.