Testing bath correlation functions for open quantum dynamics simulations

TL;DR

This work develops a rigorous, practical framework for validating approximate bath correlation functions used in open quantum dynamics simulations. By mapping complex baths to a solvable surrogate harmonic oscillator with the same BCF and employing a moment-based truncation, the authors enable exact benchmarking of thermalization observables. They compare multiple BCF fitting methods (ESPRIT, AAA, GMT&FIT, IP) across Ohmic and sub-Ohmic baths, showing that ESPRIT excels for Ohmic baths while AAA better captures low-frequency divergences in sub-Ohmic cases. Demonstrations on a two-spin system and a transmon-resonator system show that oscillator-based error estimates track qualitative trends in thermalization, guiding practical choices of $L_{\rm mod}(t)$. The framework can be extended to more complex environments and anharmonic settings, offering a scalable route to improve open quantum dynamics simulations.

Abstract

Accurate simulations of thermalization in open quantum systems require a reliable representation of the bath correlation function (BCF). Numerical approaches, such as the hierarchical equations of motion and the pseudomode method, inherently approximate the BCF using a finite set of functions, which can impact simulation accuracy. In this work, we propose a practical and rigorous testing framework to assess the validity of approximate BCFs in open quantum dynamics simulations. Our approach employs a harmonic oscillator system, where the computed dynamics can be benchmarked against known exact solutions. To enable practical testing, we make two key methodological advancements. First, we develop numerical techniques to efficiently evaluate these exact solutions across a wide range of BCFs, ensuring broad applicability. Second, we introduce a moment-based state representation that significantly simplifies computations by exploiting the Gaussian nature of the system. Applications to a two-spin system and a transmon-resonator system demonstrate that our testing procedure provides error estimates that capture the qualitative trends observed in thermalization simulations. Using this methodology, we assess the performance of recently proposed BCF construction methods, highlighting both their strengths and a notable challenge posed by sub-Ohmic spectral densities at finite temperatures.

Figures (13)

• Figure 1: Error in the model BCF from the four fitting methods. Exact BCF in the time (a) and frequency (b) domains. All quantities are shown in units of $\hbar = \omega_0 = v_0 = 1$. (c) Error $\delta L$ defined by Eq. (\ref{['eq:analysis_dL']}) as a function of $K$.
• Figure 2: Error in the steady-state expectation value, $\delta \! \braket{q^2}$, defined by Eq. (\ref{['eq:analysis_do2']}), as a function of (a) $K$ and (b) $\delta L$. In panel (b), the solid black line indicates the linear relation $\delta \! \braket{q^2} = 50 \, \delta L$.
• Figure 3: Error in the autocorrelation function. All dimensional quantities are shown in units of $\omega_0 = 1$. (a) Exact equilibrium autocorrelation function. (b) Deviation of the Fourier transform for the four fitting methods with $K = 20$.
• Figure 4: Error in the model BCF from ESPRIT and AAA with $K = 30$. All quantities are shown in units of $\hbar = \omega_0 = v_0 = 1$. (a), (b) Exact BCF in the (a) time and (b) frequency domains. (c), (d) Error in (c) ${\rm Re}[L_{\rm mod}(t)]$ and (d) $\mathcal{F}[L_{\rm mod}](\omega)$ for ESPRIT (red curve) and AAA (blue curve). In panel (c), the thin dashed vertical black line shows $t = 200$, the fitting range for ESPRIT. In panel (d), the region $|\omega| < 4 \times 10^{-4}$, where $\mathcal{F}[L](\omega) > 50$ is excluded.
• Figure 5: Comparison of the exact equilibrium expectation value (black line) with the steady-state values obtained with ESPRIT (red circles) and AAA (blue squares) as a function of $K$.