同調波轉換於介觀光學與量子力學之研究

Abstract

介觀物理是尺度介於巨觀與微觀的物理科學，使其能涵蓋兩種尺度下的物理特徵；這一尺度下的物理系統也因此孕育了不少有趣的物理現象為許多不同領域的科學家們所深深著迷。直至目前為止，相關的議題還是持續地被關注與研究。本文藉由光在量子(波動光學)與古典(幾何光學)的良好對應性，以光學系統來類比觀察介於量子力學與古典力學之間的介觀現象。再者，由於描述光學系統的波動方程式在近軸近似下與研究量子系統的薛丁格波動方程式有相當良好的數學對應性，在文中我們借助量子力學的理論完備性來探討對應於光波的量子同調態(quantum coherent states)所具有的物理特性。藉由對量子系統的深入分析，我們更容易洞悉波動光學與幾何光學之間的奧妙。也藉此更了解量子態在量子系統中所扮演的重要角色。 文中主要探討同調波在兩種光學系統中的物理面貌，包含光導管(light pipe)與球型雷射共振腔(spherical laser resonator)。看似完全不同的實驗架構，實際上卻以相同的理論架構為基礎。由量子同調態疊加的概念配合嚴謹的理論分析，疊加出來的波函數展現出坐落於古典週期性軌跡(periodic orbit)的物理圖像；透過仔細的實驗觀察，相同的空間圖像也在光學系統中被驗證。藉由同調態在光學與量子力學的相互印證之下，更確立了以量子力學為理論基石的進一步相關研究。 本文另一個重點就是透過雷射共振腔系統外的模態轉換元件(mode converter)，來連結兩群各具特色的光波同調態。而這兩群光波同調態皆具有獨特的古典週期性粒子軌道形貌，分別是利薩如(Lissajous)曲線和擺線(trochoidal)曲線。此一研究不僅以視覺化的方式呈現數學研究中拓樸學的內涵，也藉由不同耦合機制下的二維簡諧系統，具體的展現了粒子軌道的空間對應轉換關係。由於簡諧系統普遍存在於各個研究領域與問題中，空間轉換同調態的研究與實現想必會是最直接且容易的途徑來刺激或幫助解答更多不同領域中的相關問題。此外，針對實驗結果的理論分析更顯示了這些空間模態擁有很大的角動量，這對於未來的雷射技術提供了一些前瞻性的想法。 而本文另一個探討的議題在介觀物理的研究中一直扮演相當重要的角色，就是波穿透紊亂介質(disordered medium)所展現出來的局域化(localization)現象。本文藉由錐形二次諧波產生(conical second harmonic generation)的方式來觀察紊亂波函數在弱局域化(weak localization)範疇中從遍布態(extended states)到預局域化態(pre-localized states)的連續性變化；透過理論進一步分析實驗量測到的強度分布，我們成功地利用縮版的非線性σ模型(reduced version of the nonlinear sigma model)來定量地探討各種形式的強度分布其所對應不同局域化的程度，這是縮版的非線性σ模型首次在實驗上的一個應用與對照。再者，為人們所熟知的卡方分布(chi-square distributions)在此一研究中也首次被證實可以有效地使用來定量分析不同局域化的程度，且與縮版的非線性σ模型有相當良好的對應關係。由於紊亂系統的實驗並不是很容易觀察，而此一研究提供一個途徑來幫助深入了解紊亂系統所展現出來的物理圖像；另一方面，實驗結果也意味錐形二次諧波產生的方式可以協助研究紊亂晶體中複雜的結構特徵。

Mesoscopic physics, which is in between the microscopic and the macroscopic world, contains physical features of both scales. Distinctive phenomena found in the mesosopic systems give insights into the quantum-classical correspondence which has attracted lots of attention from researchers. The related issues in mesoscopic regime have been studying and paying close attention. In the thesis we employed optical systems as analog systems to investigate the connection between quantum and classical mechanics. This statement based on the good correspondence between quantum-classical mechanics and wave-ray optics. Moreover, optical wave equation was theoretically elucidated to be in the same mathematical form as the Schrödinger equation. We provided comprehensive studies for the quantum coherent states corresponding to the optical waves. With sophisticated mathematics in quantum mechanics, we are able to understand the wonderland between wave optics and ray optics and the important roles of quantum coherent states in quantum systems. Two kinds of optical systems, light pipes and a laser resonator, were discussed in the thesis. Although it seems that the two setups are totally different, they are governed by the same theoretical foundation. Within rigorous analyses, the coherent states in corresponding quantum systems revealed intriguing patterns localized on the classical periodic orbits. The same spatial patterns could be found in the optical systems. The validation of the connection between quantum and optical coherent waves enables further studies on related research based on quantum mechanics. Another topic in the thesis is the linkage of two distinctive optical coherent states localized on the periodic orbits of Lissajous and trochoidal curves. The investigation not only visualized the insight of topology in mathematics but exhibited analog transformational relationship of particle trajectories followed by different coupling mechanisms in a two-dimensional harmonic system. Hence, the realization of the converted spatial coherent states might be an accessible method for the study of fundamental science in various branches. With theoretical analyses, the coherent waves were found to carry large orbital angular momentum and might stimulate further applications. Besides the two topics mentioned on the above, another topic has been played an important role in the mesoscopic physics—the investigation of localization for disordered wave functions in random media. In this work, we obtain the disordered wave functions from the conical second harmonic generation to explore the continuous transformation of weak localization from extended to pre-localized states. We numerically verify that the experimental density distributions with different extents of weak localization can be excellently analyzed with a reduced version of the nonlinear sigma model. This is the first time that the reduced version of the nonlinear sigma model to be applied to describe the experimental results. Moreover, we perform that the chi-square distributions with fractional degrees of freedom are practically equivalent to the density distributions of the reduced version of the nonlinear sigma model. Since the observation of the disordered wave functions is not accessible, this work might provide an approach to comprehensively study the intriguing physics behind the disordered systems. On the other hand, the present results suggest the possibility of exploiting conical second harmonic generation as a diagnostic method to understand the complex topological structure of the disordered crystals.