量子與古典系統空腔態與共振態之研究：驅動源與耦合強度之影響

Abstract

實驗上波動系統的共振頻譜及共振模態常被假定可由理論計算荷姆霍茲方程式(Helmholtz Equation)在相同侷限邊界下所得的本徵值及本徵態直接描述。然而，真實系統中驅動源與共振腔體間的耦合作用無可避免地造成共振行為與理想本徵系統特質的偏差。在此論文中，我們針對量子彈子球檯、微共振腔雷射、聲學平板振盪以及RLC電路網絡等具有與散射位勢或驅動源耦合作用的量子和古典系統進行研究。在弱耦合作用範疇中，我們發現微共振腔雷射實驗上所量測的共振頻譜仍可用以顯現理論上理想侷限系統的本徵頻譜行為。另一方面，對於振盪平板和RLC電路網絡等具有與驅動源強耦合作用的共振系統，其共振頻譜及共振模態均呈現與空腔時的本徵頻譜及本徵態有非常顯著的偏差。根據這些實驗發現，我們進一步發展一套理論模式來分析驅動源耦合作用下的共振腔行為。我們證實這些由理論模式推導出的共振模態可成功描述弱耦合作用至強耦合作用下的實驗觀察。

It is commonly presumed that the resonant spectrum and resonant mode in experimental wave systems can be straightforwardly described by the theoretical eigenvalue and eigenmode solved by the homogeneous Helmholtz equation in the same bounded domain. However, the coupling between the excitation source and the cavity in experimental resonator inevitably deviate the resonant behaviors from the intrinsic properties of ideal eigen-system. In this thesis, several quantum and classical systems including quantum billiards, microcavity lasers, acoustic vibrating plates, and RLC circuit networks with mutual coupling to a scattering potential or an excitation source are explored. In the weak-coupling regime, it is confirmed that the experimental resonant spectra of the microcavity laser can still be utilized to manifest the characteristics of theoretical eigenvalue spectra of the ideally bound system. On the other hand, for the vibrating plates and RLC circuit networks in which the resonators are strongly coupled with the excitation source, the resonant spectra and resonant modes are found to be dramatically different from the eigenvalues and eigenmodes of the source-free cavity. According to these experimental findings, we further develop a theoretical model for analyzing resonators subject to the coupling from excitation source. The resonant modes derived from the theoretical model are verified to successfully describe the experimental observations from weak to strong coupling regime.