Taming Quantum Noise for Efficient Low Temperature Simulations of Open Quantum Systems

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.

Figure 1

Matsubara poles of the spectral noise power $S_{\beta }(\omega )$ of a thermal reservoir at $T=0$, where they merge into a branch cut along the imaginary axis (blue) together with the optimized pole distribution (red) based on its rational decomposition [Eq. (4)]. For the latter, data refer to a subohmic spectral density [Eq. (7)] with $s=1/2$, $\alpha =0.05$, and ${\omega }_c=20$ (in arbitrary units) with an accuracy $\delta ={10}^{-8}$ in the domain $\mathcal{A}=[-{10}^3,-{10}^{-4}]\cup [{10}^{-4},{10}^3]$ [63]. Inset: low-frequency range.

Figure 2

Validation of the Shiba relation [Eq. (10)] of the subohmic spin-boson model [Eq. (7)] with $\epsilon =0$, $\mathrm{\Delta }=1$ (in arbitrary units), and $s=1/2$ in the long time domain. Inset: long time behavior on a logarithmic scale. Other parameters are as in Fig. 1.

Figure 3

Relaxation dynamics of the subohmic spin-boson model for various spectral exponents $s$ at $T=0$ displaying the transition from a delocalized ($⟨{\sigma }_z⟩=0$) to a localized ($⟨{\sigma }_z⟩\ne 0$) asymptotic state. Parameters are as in Figs. 1 and 2.

Figure 4

Long-time dynamics of the subohmic spin-boson model with $s=1/2$ at $T=0$ and strong coupling $\alpha =0.4$ as predicted by TD-DMRG (black, 1000 modes), ML-MCTDH (green, 1000 modes), and FP-HEOM (red, 21 FP-modes; accuracy corresponds to line width). On short timescales, all approaches coincide (inset). Parameters are as in Figs. 1 and 2.