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Title: An exact formulation of the time-ordered exponential using path-sums

Abstract

We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.

Authors:
 [1];  [2];  [3];  [1];  [4]
  1. Clarendon Laboratory, Department of Physics, University of Oxford, Oxford OX1 3PU (United Kingdom)
  2. Colby College, 4000 Mayflower Hill Dr., Waterville, Maine 04901 (United States)
  3. Department of Physics and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 Munich (Germany)
  4. (Singapore)
Publication Date:
OSTI Identifier:
22403149
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 56; Journal Issue: 5; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONTINUED FRACTIONS; DIAGRAMS; GRAPH THEORY; MATRICES; TIME DEPENDENCE

Citation Formats

Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk, Lui, K., Thwaite, S. J., Jaksch, D., and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543. An exact formulation of the time-ordered exponential using path-sums. United States: N. p., 2015. Web. doi:10.1063/1.4920925.
Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk, Lui, K., Thwaite, S. J., Jaksch, D., & Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543. An exact formulation of the time-ordered exponential using path-sums. United States. doi:10.1063/1.4920925.
Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk, Lui, K., Thwaite, S. J., Jaksch, D., and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543. Fri . "An exact formulation of the time-ordered exponential using path-sums". United States. doi:10.1063/1.4920925.
@article{osti_22403149,
title = {An exact formulation of the time-ordered exponential using path-sums},
author = {Giscard, P.-L., E-mail: p.giscard1@physics.ox.ac.uk and Lui, K. and Thwaite, S. J. and Jaksch, D. and Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543},
abstractNote = {We present the path-sum formulation for the time-ordered exponential of a time-dependent matrix. The path-sum formulation gives the time-ordered exponential as a branched continued fraction of finite depth and breadth. The terms of the path-sum have an elementary interpretation as self-avoiding walks and self-avoiding polygons on a graph. Our result is based on a representation of the time-ordered exponential as the inverse of an operator, the mapping of this inverse to sums of walks on a graphs, and the algebraic structure of sets of walks. We give examples demonstrating our approach. We establish a super-exponential decay bound for the magnitude of the entries of the time-ordered exponential of sparse matrices. We give explicit results for matrices with commonly encountered sparse structures.},
doi = {10.1063/1.4920925},
journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 5,
volume = 56,
place = {United States},
year = {2015},
month = {5}
}