從微觀角度出發的非馬可夫薛丁格—朗之萬方程式

Abstract

我們從微觀角度推導了一個包含非馬可夫動力學 (non-Markovain dynamics) 的量子朗之萬方程式 (quantum Langevin equation)，並使用譜密度 (spectral density) 來描述環境影響。我們將此量子朗之萬方程式加入含時薛丁格方程式，成為薛丁格——朗之萬方程式以模擬波函數的動力學。在此之上我們探究了此波函數動力學能夠具有馬可夫性質的參數範圍。為此，我們定義了馬可夫係數 (Markovian coefficient) 並用於薛丁格—朗之萬方程式的馬可夫近似。我們的結果顯示在德魯德—勞侖茲譜密度環境下，當系統頻率低且耦合 (coupling) 強度低的情況馬可夫近似是合適的。但以超歐姆 (super-Ohmic) 譜密度環境下馬可夫近似在低頻表現不佳，但在中等頻率下表現較好。而耦合強度幾乎不影響此近似。

We present a derivation of a quantum Langevin equation (QLE) from a microscopic perspective. This approach includes non-Markovian dynamics, and describes the environmental effects using the spectral density based on the microscopic model. We further simulate the wavefunction dynamics based on the QLE, leading to a Schrödinger-Langevin equation (SLE). Furthermore, we investigate the regime where the dissipative dynamics can be considered Markovian. To this end, we define a Markovian coefficient and utilize it in the Markovian approximation of the Schrödinger-Langevin equation. Our results demonstrate that for the Drude-Lorentz bath, the Markovian approximation is accurate in the regime characterized by low system frequency and weak system-bath coupling; however, for the super-Ohmic bath, the Markovian approximation fails at low and high system frequency, but is reliable in the intermediate system frequency regime. The damping coefficient does not affect the approximation at intermediate system frequency apparently.