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 delocalisation of the bath modes. This delocalisation can be quantified by the radius of gyration squared {mathcal R}²(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, {mathcal R}²(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 {mathcal R}<sup>2</sup>(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 {mathcal R}²(ω) to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.