Exploiting the path-integral radius of gyration in open quantum dynamics

A major challenge in open quantum dynamics is the inclusion of Matsubara-decay terms in the memory kernel, which arise from the quantum-Boltzmann delocalization of the bath modes. This delocalization can be quantified by the radius of gyration squared R2(ω) of the imaginary-time Feynman paths of the bath modes as a function of the frequency ω. In a hierarchical equations of motion (HEOM) calculation with a Debye-Drude spectral density, R2(ω) is the only quantity that is treated approximately (assuming convergence with respect to hierarchy depth). Here, we show that the well-known Ishizaki-Tanimura correction is equivalent to separating smooth from "Brownian" contributions to R2(ω) and that modifying the correction leads to a more efficient HEOM in the case of fast baths. We also develop a simple "A4" adaptation of the "AAA" (adaptive Antoulas-Anderson) algorithm in order to fit R2(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.