TY - JOUR
T1 - Removing instabilities in the hierarchical equations of motion
T2 - Exact and approximate projection approaches
AU - Dunn, Ian S.
AU - Tempelaar, Roel
AU - Reichman, David R.
N1 - Funding Information:
The authors thank Gregory Beylkin, Seogjoo Jang, Ramin Khajeh, Benedikt Kloss, Matthew Reuter, and Qiang Shi for helpful and enlightening discussions. I.S.D. acknowledges support from the United States Department of Energy through the Computational Sciences Graduate Fellowship (DOE CSGF) under Grant No. DE-FG02-97ER25308. D.R.R. acknowledges funding from NSF Grant No. CHE-1839464.
Funding Information:
The authors thank Gregory Beylkin, Seogjoo Jang, Ramin Kha-jeh, Benedikt Kloss, Matthew Reuter, and Qiang Shi for helpful and enlightening discussions. I.S.D. acknowledges support from the United States Department of Energy through the Computational Sciences Graduate Fellowship (DOE CSGF) under Grant No. DE-FG02-97ER25308. D.R.R. acknowledges funding from NSF Grant No. CHE-1839464.
Publisher Copyright:
© 2019 Author(s).
PY - 2019/5/14
Y1 - 2019/5/14
N2 - 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. We also discuss issues associated with the non-normality of HEOM.
AB - 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. We also discuss issues associated with the non-normality of HEOM.
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U2 - 10.1063/1.5092616
DO - 10.1063/1.5092616
M3 - Article
C2 - 31091920
AN - SCOPUS:85065657741
SN - 0021-9606
VL - 150
JO - Journal of Chemical Physics
JF - Journal of Chemical Physics
IS - 18
M1 - 184109
ER -