The hierarchical equations of motion (HEOM), derived from the exact Feynman-Vernon path integral, is one of the most powerful numerical methods to simulate the dynamics of open quantum systems. Its applicability has so far been limited to specific forms of spectral reservoir distributions and relatively elevated temperatures. Here we solve this problem and introduce an effective treatment of quantum noise in frequency space by systematically clustering higher order Matsubara poles, equivalent to an optimized rational decomposition. This leads to an elegant extension of the HEOM to arbitrary temperatures and very general reservoirs in combination with efficiency, high accuracy, and long-time stability. Moreover, the technique can directly be implemented in other approaches such as Green’s function, stochastic, and pseudomode formulations. As one highly nontrivial application, for the subohmic spin-boson model at vanishing temperature the Shiba relation is quantitatively verified which predicts the long-time decay of correlation functions.

中文翻译：

---

为开放量子系统的高效低温模拟驯服量子噪声

分层运动方程 (HEOM) 源自精确的 Feynman-Vernon 路径积分，是模拟开放量子系统动力学的最强大的数值方法之一。迄今为止，它的适用性仅限于特定形式的光谱储层分布和相对升高的温度。在这里，我们解决了这个问题，并通过系统地聚类高阶 Matsubara 极点来引入频率空间中量子噪声的有效处理，相当于优化的有理分解。这导致 HEOM 优雅地扩展到任意温度和非常通用的储层，并结合了效率、高精度和长期稳定性。此外，该技术可以直接在其他方法中实现，例如格林函数、随机和伪模公式。