Exact and Approximate Projection Approaches for Removing Instabilities From the Hierarchical Equations of Motion
Ian Dunn, Columbia University
The hierarchical equations of motion method (HEOM) is a numerically exact approach to computing the reduced dynamics of an open quantum system in contact with a bath. In HEOM, one converges upon the exact dynamics by successively increasing the number of coupled equations of motion (hierarchy layers). We have noticed that when the bath consists of a finite number of undamped harmonic oscillators, HEOM contains instabilities that grow exponentially in time. At great cost, these instabilities may be delayed to later times by increasing the number of hierarchy layers. We show that these instabilities also can be completely removed via projection onto the space of stable eigenmodes, and we find that at least in small systems the remaining projected dynamics computed with few hierarchy layers are quite close to the exact reduced dynamics that require a large number of hierarchy layers. Recognizing that computation of the eigenmodes might be prohibitive, e.g. for larger systems, we present an approximate 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 both the exactly and approximately projected HEOM.
Abstract Author(s): Ian S. Dunn, Roel Tempelaar, David R. Reichman