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 ${\mathcal{R}}^2(\omega )$ of the imaginary-time Feynman paths of the bath modes as a function of the frequency $\omega$. In a Hierarchical Equations of Motion (HEOM) calculation with a Debye–Drude spectral density, ${\mathcal{R}}^2(\omega )$ 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 ${\mathcal R}^2(ω)$, 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 ${\mathcal{R}}^2(\omega )$ to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.