在量子尺度下光與物質的相互作用，向來是研究者所關注的焦點，因其常具有反直覺的量子特性。對涉及光與物質交互作用的開放性量子系統研究，為探索量子力學的基礎提供了一個可行的研究平台。開放性量子系統理論儼然成為認識與深入探索量子現象的最有力工具之一。然而開放性量子系統問題的研究，通常奠基於玻恩-馬可夫近似，這個近似方法通常需要假設環境是很“大”的，從而使得環境可以影響量子系統但卻不受系統所影響。

在本論文中，我們考慮玻恩-馬可夫近似的情況之下，選擇量子點來研究開放量子系統的動態行為，該量子點可以透過實驗設計形成二能階系統，進一步耦合到金屬奈米粒子。而光在金屬奈米粒子上所誘導出的表面電漿子，可以將光能轉化為電能，克服光的繞射極限，進而顯著增強局部的電場。利用電漿子的優勢，我們能更深入了解與表面電漿子耦合的量子點系統其動力學特徵。此外，表面電漿子耦合量子點系統的動力學，可以透過林德布拉德主方程所推導出的有效非厄密特哈密頓來描述。其中，非厄密系統所產生的例外點在提高量子感測器的靈敏度上深具潛力。透過量子電動力學的方法，我們藉由觀測表面電漿子耦合量子點系統實部和虛部的本徵能譜，可以得知在何種表面電漿子耦合的強度下，能使得該系統產生我們想觀測到的例外點。我們也比較了單一量子點和數個量子點耦合到表面電漿子對系統例外點的影響，進一步發現，相較於單一個量子點，多個量子點同時耦合到表面電漿子的集體效應所產生的例外點，在集體效應的幫助下，其產生條件不僅不需要像單一量子點所需和表面電漿子如此大的耦合強度，而且根據訊噪比的分析，我們發現其例外點較為穩定而且更容易被量子點的光譜所偵測到。

此外，為了研究有別於上述之弱交互作用的系統，我們進一步屏除了玻恩-馬可夫近似，以研究強相關的量子系統，亦即和環境強相互作用的系統。我們考慮量子點和電極強耦合的系統，這種量子點-電極強耦合系統可以表現出多體效應，即近藤效應-由於量子點內的電子和其周圍的電子產生多體的量子糾纏所引起。近藤效應的有效調控被視為是未來自旋電子器件的關鍵技術之一，因此，我們提出了一種將量子點進一步超強耦合到共振腔的方法，藉由共振腔引起的電子光子交互作用來控制整個系統的近藤效應。

然而在理論計算上，量子系統的近藤效應是無法以玻恩-馬可夫近似的主方程來模擬的。因此我們採用分層運動方程 (HEOM)，即一種數值精確的方法，除了可以刻畫量子點態密度中出現的近藤共振現象，也可以適用於任何複雜的量子點系統，如本研究中所考慮的量子點-共振腔耦合系統。在這個基礎上，我們運用分層運動方程所數值模擬出來之量子點態密度結果，來定性分析該量子點-共振腔耦合系統的近藤效應，如何受電子和超強光子相互作用所影響。我們先計算單能階量子點耦合共振腔系統的量子點態密度的結果，來討論分層運動方程在不同參數下的收斂程度，給出了可行的參數範圍。此外，我們也考慮二能階的量子點耦合到共振腔的系統，歸納出量子點與共振腔的耦合可以分為兩種形式，即縱向耦合和橫向耦合。從量子點態密度的結果中我們發現，共振腔對量子點的超強縱向耦合會使兩邊的Hubbard peaks同時往低頻的方向平移。而當我們將量子點與共振腔的耦合模式從超強縱向耦合調整至超強橫向耦合，我們發現到共振腔的耦合可以顯著抑制近藤共振。這種近藤效應的抑制可以從量子點態密度譜中得知，由位於費米能量周圍的電子被激發參與電子-光子裹態的形成，進而部分破壞由量子點和環境電子所組成的多體單重態。

Research on light-matter interaction has always received great attention due to its counter-intuitive quantum properties. The investigation of open quantum systems involving light-matter interaction provides a viable platform for exploring the fundamentals of quantum mechanics. In most cases, the study of the open system problem is based on the Born-Markov approximation, for which the coupling environment is assumed to be "large", so that it can affect the quantum system but is unaffected by the system.

With the help of the Born-Markov approximation, we investigate the dynamics of an open quantum system: the point-like quantum dots (QDs), which can be properly designed to serve as a two-level system coupled to the plasmonic nanoparticle. The surface plasmon induced by the light on the metal nanoparticle can convert light energy into electrical energy, overcoming the diffraction limit of light, and enhancing the local electric field in the near-field. Taking advantage of the above features, the controlled dynamics of QDs strongly coupled to surface plasmon polariton (SPP) can be characterized by effective non-Hermitian Hamiltonians in the frame of the Lindblad master equation. The quantum exceptional points of non-Hermitian Hamiltonian enable such an SPP–QD system to become a promising candidate for quantum sensing. By using an analytical quantum electrodynamics approach, exceptional points are manifested as a result of a strong-coupling effect and observable in a drastic splitting of originally coalescent eigenenergies. We show that exceptional points can also occur when several quantum emitters are collectively coupled to the dipole mode of localized surface plasmons. Such a quantum collective effect not only relaxes the strong-coupling requirement for an individual QD, but also results in a more stable generation of the exceptional points. Furthermore, we point out that the exceptional points can be explicitly revealed in the power spectra. A generalized signal-to-noise ratio, accounting for both the frequency splitting in the power spectrum and the system's dissipation, evidently shows that a collection of quantum emitters coupled to a nanoparticle provides better performance of detecting exceptional points, compared to that of a single quantum emitter.

Moreover, we also go beyond the Markov approximation in order to study strongly correlated quantum systems, such as the QD strongly coupled to leads. This quantum dot-lead hybrid system can exhibit the many-body effect, namely, the Kondo effect, caused by the entanglement between the electrons in leads and QD due to the strong system-bath coupling. The manipulation of the Kondo effect is one of the critical abilities for the development of future spintronic devices. We propose an alternative way to control the Kondo effect by using the additional electron-photon interactions between a single-mode cavity and QD.

The Kondo effect cannot be simulated under the Markovian quantum master equation. As the dot-lead interactions cannot be treated perturbatively, we employ the hierarchical equations of motion (HEOM) approach, i.e., one of the numerically exact methods, not only because it can capture the Kondo resonance emerging in the density of states of the QD, but for the reason that it can apply to any complex QD system with only slight derivation effort. Within the HEOM scheme, we provide a qualitative description of how the Kondo effect is affected by electron-photon interaction in the ultra-strong coupling regime. The degree of convergence in the results of DOSs, obtained by HEOM, have been discussed. Due to the controllability of the different types of cavity coupling, i.e., ultra-strong longitudinal or transverse coupling in the QD-cavity system, we show that when the QD is tuned from single-level to two-level, longitudinal electron-photon interaction in the ultrastrong coupling regime results in the shift of both Hubbard bands in analogy to the case of single-level QD. In addition, the Kondo resonance is noticeably suppressed by changing the longitudinal type to the transverse type of ultrastrong coupling to the cavity. The suppression of the Kondo resonance is caused by the formation of the electron-photon dressed states, which require the electrons around the Fermi energy to participate, thereby partially destructing the many-body singlet states.

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