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Title: Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches

Abstract

The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. Finally, we discuss issues associated with the non-normality of HEOM.

Authors:
ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [1]
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  1. Columbia Univ., New York, NY (United States)
Publication Date:
Research Org.:
Krell Institute, Ames, IA (United States)
Sponsoring Org.:
USDOE Office of Science (SC); National Science Foundation (NSF)
OSTI Identifier:
1610347
Alternate Identifier(s):
OSTI ID: 1512551
Grant/Contract Number:  
FG02-97ER25308; CHE-1839464
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 150; Journal Issue: 18; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CHEMISTRY; Chemistry; Physics

Citation Formats

Dunn, Ian S., Tempelaar, Roel, and Reichman, David R. Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches. United States: N. p., 2019. Web. doi:10.1063/1.5092616.
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Dunn, Ian S., Tempelaar, Roel, & Reichman, David R. Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches. United States. https://doi.org/10.1063/1.5092616
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Dunn, Ian S., Tempelaar, Roel, and Reichman, David R. Fri . "Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches". United States. https://doi.org/10.1063/1.5092616. https://www.osti.gov/servlets/purl/1610347.
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@article{osti_1610347,
title = {Removing instabilities in the hierarchical equations of motion: Exact and approximate projection approaches},
author = {Dunn, Ian S. and Tempelaar, Roel and Reichman, David R.},
abstractNote = {The hierarchical equations of motion (HEOM) provide a numerically exact approach for computing the reduced dynamics of a quantum system linearly coupled to a bath. We have found that HEOM contains temperature-dependent instabilities that grow exponentially in time. In the case of continuous-bath models, these instabilities may be delayed to later times by increasing the hierarchy dimension; however, for systems coupled to discrete, nondispersive modes, increasing the hierarchy dimension does little to alleviate the problem. We show that these instabilities can also be removed completely at a potentially much lower cost via projection onto the space of stable eigenmodes; furthermore, we find that for discrete-bath models at zero temperature, the remaining projected dynamics computed with few hierarchy levels are essentially identical to the exact dynamics that otherwise might require an intractably large number of hierarchy levels for convergence. Recognizing that computation of the eigenmodes might be prohibitive, e.g., for large or strongly coupled models, we present a Prony filtration algorithm that may be useful as an alternative for accomplishing this projection when diagonalization is too costly. We present results demonstrating the efficacy of HEOM projected via diagonalization and Prony filtration. Finally, we discuss issues associated with the non-normality of HEOM.},
doi = {10.1063/1.5092616},
journal = {Journal of Chemical Physics},
number = 18,
volume = 150,
place = {United States},
year = {Fri May 10 00:00:00 EDT 2019},
month = {Fri May 10 00:00:00 EDT 2019}
}
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