非阿貝爾介觀系統中的量子相位與糾纏

Abstract

量子相位和糾纏是量子力學中兩個重要的主題，其相應領域有廣泛的研究。然而，對於這兩個主題的結合研究相對較少。在這篇論文中，我們以納米尺度系統作為模型，探索各種理論問題。首先，我們介紹了傳統的傳輸理論庫博公式和非平衡格林函數，以建立器件中的電導和力的概念。然後，我們深入討論了規範理論，這提供了一種研究力的替代方法。與經典物理不同，規範理論在量子力學中可以發展成非阿貝爾規範理論。規範理論提供了一種研究量子相位的有效方法，我們將介紹著名的概念，如貝瑞相位和阿哈羅諾夫-布姆效應（AB效應），以及阿哈羅諾夫-卡舍爾效應（AC效應）。通過用電子自旋替換AC效應中的磁圈，我們獲得了非阿貝爾AC相位。我們將糾纏自旋對輸入到量子方形環中，觀察到纏結強度對AC相位的生成和湮滅的影響。在最大糾纏的情況下，系統的動態相位完全消失，幾何相位變成離散的。相反，在部分糾纏和非糾纏的情況下，幾何相位和動態相位取決於環的波長和尺寸，從而產生離散或局部連續的值。我們已經證明了這種系統中的量子糾纏極大地簡化了對幾何相位研究的未來實驗努力。最後一個主題是關於2D器件中曲面效應。當粒子被限制在曲面或曲線上運動時，哈密頓量中的動能項將有一個額外位能修正項。此外，自旋-軌道耦合項將具有與曲面相關的規範，這將影響自旋-軌道轉矩並產生許多未來的研究機會。

Quantum phase and entanglement are both important topics in quantum mechanics, with extensive research in their respective fields. However, there is relatively little discussion on their combined study. In this thesis, we use nanoscale systems as our model to explore various theoretical topics. We first introduce the classical transportation theory Kubo formula and non-equilibrium Green's function to establish the concept of conductivity and force in the device. We then delve into the discussion of gauge theory, which provides an alternative approach to studying force. Unlike classical physics, gauge theory can be advanced into non-Abelian gauge theory in quantum mechanics. Gauge theory offers an effective method for studying quantum phase, and we will introduce well-known concepts such as Berry''s phase and the Aharonov-Bohm (AB) effect, as well as the Aharonov-Casher (AC) effect. By replacing the magnetic coil in the AC effect with electron spin, we obtain the non-Abelian AC phase. We design the input of entangled spin pairs into a quantum square ring and observe that the strength of entanglement affects the generation and annihilation of the AC phase. In the case of maximal entanglement, the system's dynamic phase is completely eliminated, and the geometric phase becomes discrete. In contrast, in the cases of partial entanglement and non-entanglement, the geometric and dynamic phases depend on the wavelength and size of the ring, resulting in either discrete or locally continuous values. We have demonstrated that quantum entanglement in this system significantly facilitates future experimental endeavors in investigating geometric phases. The last topic is about the effects of curved surfaces in 2D devices. When particles are confined to move on curved surfaces or paths, the kinetic energy term in the Hamiltonian will have an additional potential correction term. Additionally, the spin-orbit coupling term will have a curved-related gauge, which will affect the spin-orbit torque and give rise to many future research opportunities.