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

TL;DR

This work reframes Matsubara-decay terms in open quantum dynamics through the lens of the bath radius of gyration $\mathcal{R}^2(\omega)$, showing that the Ishizaki--Tanimura correction effectively isolates smooth and Brownian components to accelerate HEOM for fast baths. A modified IT correction (mIT) further improves convergence by replacing the high-frequency variance with a constant, reducing unphysical bath-parameter dependencies. The authors introduce an A4 adaptation of the AAA algorithm to fit $\mathcal{R}^2(\omega)$ as a sum over purely imaginary poles, enabling an efficient pole-based representation that dramatically speeds up low-temperature HEOM for Debye--Drude baths compared with Padé-based approaches. The method holds promise for broader applicability to other spectral densities and even Fermionic baths, offering a practical route to scalable, accurate simulations of open quantum systems at cryogenic temperatures.

Abstract

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 ${\mathcal R}^2(ω)$, 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 ${\mathcal R}^2(ω)$ to a sum over poles, which results in an extremely efficient implementation of the standard HEOM method at low temperatures.

Figures (8)

• Figure 1: Comparison of the Matsubara and ring-polymer expansions of ${\mathcal{R}}^2(\omega)$ (Eqs. (\ref{['rexpa']}) and (\ref{['polly']})), truncated at $\overline M=\overline P=K=10$ terms, with the Ishizaki--Tanimura (IT)-corrected Matsubara expansion (Eq. (\ref{['IT-rgapprox']})), as a function of the bath frequency $\omega$ (for $\beta=1$).
• Figure 2: Convergence of the truncated ring-polymer and IT-corrected-Matsubara approximations to ${\mathcal{R}}^2(\omega)$, and of the resulting HEOM calculations of $\langle \hat{\sigma}_z(t) \rangle$, as a function of the number of poles $K$. The HEOM calculations were carried out for the spin-boson system of Eq. (\ref{['sob']}), with $\epsilon=0,\Delta=1,\gamma=1,\eta=1$, at $\beta=8$.
• Figure 3: Decomposition of an imaginary-time Feynman path $x(\tau)$ for a bath mode of frequency $\omega=1$, into smooth $n\le K$ and 'Brownian' $n>K$ Matsubara components, for $K=10$ and $\beta=1$. Note the low amplitude of the Brownian path in comparison with the classical variance (=1).
• Figure 4: Demonstration that the modified version of the IT correction term (Eq. (\ref{['Kmart']})) gives cleaner convergence for a fast bath than the original form (Eq. (\ref{['Kold']})). All parameters are the same as in Fig. 2 except that $\gamma=61.3$ and $\beta=1$.
• Figure 5: Convergence of the IT-corrected Matsubara, [N/N] Padé, and A4 approximations to $\mathcal{R}^2(\omega)$ and of the resulting HEOM calculations, with respect to the number of terms $K$ in the sum-over-poles expansion. The system and bath parameters are the same as in Fig. 2 and $\beta=50$.