具有有效熱振子模型框架的層次運動方程

Abstract

層次運動方程式（Hierarchical Equations of Motion, HEOM）方法可在諧振環境下對開放量子系統動力學進行數值精確模擬，然而在強系統-環境耦合或極低溫條件下，HEOM 方法的計算成本將變得難以負擔。此外，HEOM 模擬中所採用的環境頻譜密度（spectral density）常受限於 Debye–Lorentz 形式，以利時間相關函數（Time Correlation Function, TCF）的指數展開。為克服這些限制，我們提出一種有效熱振盪模型（Effective Thermal Oscillator Model, ETOM），此方法可直接將 TCF 解釋為一系列振盪性的指數衰退項，避免了低溫修正的繁瑣處理，並顯著減少在模擬任意頻譜密度時所需的輔助密度算符（Auxiliary Density Operators, ADO）數量。此外，我們亦結合 GPU 加速以提升計算效率，使 HEOM 方法能更快速地模擬開放量子系統的動力學。針對一系列模型系統，我們應用 ETOM-HEOM 方法進行族群動力學與二維電子光譜（2D electronic spectra）模擬並分析光譜峰值訊號變化帶來的非馬可夫性質。最後，我們將 ETOM-HEOM 與標準 HEOM 及其他數值精確方法進行比較，驗證新方法之正確性與可靠性。我們期望 ETOM-HEOM 能成為一套功能強大且開放原始碼的模擬工具，為化學與物理領域中重要的量子動力學現象提供高效且精確的模擬工具。

Hierarchical equations of motion (HEOM) approach offers exact numerical simulations of open quantum system dynamics under harmonic baths, but it becomes computationally intractable for strong system-bath coupling or low temperature regime. Furthermore, the bath spectral density used in HEOM simulations is often constrained to the Debye–Lorentz form for time correlation function (TCF) exponential expansion. To overcome these limitations, we propose an effective thermal oscillator model (ETOM), which directly interprets TCF to a series of oscillatory exponential terms and avoids both cumbersome low-temperature corrections and large number of auxiliary density operators needed to simulate arbitrary spectral densities. Moreover, we also incorporate GPU acceleration to enhance computational efficiency, enabling faster simulations of open quantum system dynamics using the HEOM approach. Applications of the ETOM-HEOM method for calculations of population dynamics as well as 2D electronic spectra for a series of model systems across a wide parameter regime were carried out to analyze the variance in 2D peaks signal and explore the non-Markovian property of the system. Finally, we compared ETOM-HEOM with standard HEOM and other numerically exact method to confirm the validity of the new approach. We expect ETOM-HEOM to become a powerful and open source tools for efficient and accurate simulations of quantum dynamical phenomena important in chemistry and physics.