光合作用複合物在常溫下的長存性量子同調性已被二維光子回聲(2DPE)實驗證實，而可以保有常溫下的長存性量子同調性的來源已引起了廣泛的討論。我們在本論文中提出了一個非馬可夫性的二聚物模型來模擬光合作用複合物的動力學，這個二聚物模型不需透過任何特定的數值計算技巧便可得到解析解。我們的理論認為，二聚物系統與其蛋白質環境中某些震動模態的非馬可夫耦合，在延續量子同調性的機制中扮演著重要的角色。這樣的機制可以簡單的類比成量子光學中常見的外加驅動Rabi震盪。
接著，我們簡化了該二聚物模型成為一個自旋玻色子模型，並且運用了兩種不同的微擾法推導得到兩種不同的主方程式(TC2, TL2)。為了瞭解這兩種近似的方程式離精確解的偏差多大，我們把這兩種微擾主方程式與階層式運動方程式(HEOM)作比較，因為階層式運動方程式被認為可以得到數值精確的解。我們發現到，在弱交互作用情況下，兩種微擾主方程式都可以得到良好的近似解。然而，當系統與環境的交互作用強度逐漸增強，到達中介交互作用的情況下，兩種微擾主方程式各自會顯現出不同的缺點。TC2會高估系統的量子同調性而TL2會高估消相干率。
為了能夠量化以上各個方程式的記憶效應，我們接下來簡單的回顧幾種非馬可夫測度。由於這個非馬可夫測度的問題在開量子系統理論中具有相當的重要性，所以這個領域近來引起了人們的興趣。我們運用了RHP測渡法量化我們的光合作用二聚物模型中的非馬可夫性，藉此我們可以發現長存性的量子同調性與非馬可夫性之間的關聯。另一個更進一步的應用為，我們比較了我們的各種自旋波色子模型的非馬可夫性，探討非馬可夫性與描述系統動力學各個參數之間的關聯。此外我們注意到一個違反直覺的現象， 在我們的自旋波色子模型中，非馬可夫性會隨著環境溫度與交互作用強度的增加而增加。這樣的工作也提示了我們非馬可夫測度理論可以當作近似方程式的的一種效能指標，指示該近似方程式對於記憶效應的描述是否恰當。
為了能夠更加透徹的瞭解非馬可夫測度理論，我們運用了k-正定映像理論將非馬可夫測度加以推廣，得到了階層式非馬可夫性。此外我們也發展出了一套工具，我們稱之為k-可分割性相位圖，這樣的k-可分割性相位圖可將階層式非馬可夫性進一步的視覺化，並且提供一個一致的平台來探討比較各種被提出的非馬可夫測度。藉著k-可分割性相位圖來進行各種非馬可夫測度的比較，可以顯示出各種測度不一致的原因，以及更加深入的瞭解各種測度的本質。

The long-lived quantum coherence in photosynthetic complexes existing in ambient temperature has been reported experimentally by two-dimensional photon echo (2DPE) technique and the
origin of this phenomenon has attracted vigorous debates. We propose an analytical non-Markovian dimer model without invoking any specific numerical implementation to simulate the
dynamics of the photosynthetic complexes. Our theory suggests that the non-Markovian coupling to the vibronic motion of the surrounding molecules plays an important role in sustaining
the quantum coherence. The mechanism is analogous to an externally driven Rabi-oscillations in quantum optics.

Sequentially, we simplify the dimer model to a spin-boson model and apply two perturbative approaches to evaluate different master equations. To understand how the two approximative
equations deviate form the exact results, we compare those with the hierarchy equations of motion (HEOM) method, which is considered to be numerically exact. We find out that both
approximations perform well in the weak-coupling regime. However, when it goes to the intermediate-coupling regime, the two approximations show different drawbacks. The second-order
time-convolution (TC2) model will over-estimate the quantum coherence, whereas, the second-order time-local (TL2) model over-estimates the decoherence rates
and leads to a sluggish dynamics.

To quantify the effect of past memory in the above equations, we briefly introduce several measures of non-Markovianity. This topic has recently attracted great interests due to its
fundamental importance in the theory of open quantum system. We apply the RHP measure to quantify the non-Markovianity of our photosynthetic dimer model and find out the
relations between the long-lived quantum coherence and non-Markovianity. For further applications, we compare the non-Markovianity of three approaches (TC2, TL2 and
HEOM) and investigate its dependance on several parameters describing the system. We find out a counter-intuitive increase of non-Markovianity with both temperature and with the coupling
strength to the environment. This suggests that the measure of non-Markovianity acts as a benchmark to quantify how well an approximative equations takes the memory effects into account.

To acquire a more fine-grained insight of non-Markovianity, we further adopt the theory of $k$-positive maps to generalize a hierarchy of non-Markovianity. We also develop a useful tool,
$k$-divisibility phase diagram, to visualize the hierarchy of non-Markovianity. This phase diagram provides a unified framework to study distinct measures of
non-Markovianity and show correlations among them. Comparison of those measures of non-Markovianity by means of $k$-divisibility phase diagram shows the origin of discrepancy and the
circumstance under which these measures get reconciliation.

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