光激子是一種半光、半物質的準粒子,在激子和光子的強交互作用下,可以在半導體二維度的量子井微共振腔內產生。因為光子的輕質量搭配激子的交互作用力,半導體共振腔內的光激子在臨界激發條件以上可以透過受激發累積效應佔據到一個較低的量子凝聚態。這個現象可以經由觀察空間或時間同調性、量子化渦旋、激發能譜或是超流特性等等加以印證。因此研究光激子的動力學行為, 探討其物理特性以供實驗佐證是不可或缺的一環。本篇論文使用複數形式的Gross-Pitaevskii方程式來理論研究利用有限大小光束激發的光激子凝聚態。我們使用兩種計算模型,一種是利用單一微分方程式來描述凝聚態的動態行為,其中激發光束所帶來的增益及凝聚態的損耗被整合在傳統冷原子系統所使用的傳統Gross-Pitaevskii方程式當中;另外一種是考慮凝聚態的增益來自於周遭環境的熱庫,此熱庫本身有鬆弛行為,而且能持續提供凝聚態增益的粒子數。我們先計算凝聚態在諧和式函數位能侷限狀態下的動態穩定解;隨之透過Bogoliubov動態穩定性分析探討其穩定性,得到一個對應激發參數,也就是激發光束大小及激發強度的相圖。在這個架構下,我們可以進行一些像是拉力、臨界流速等超流特性研究,量子化渦旋,BKT 相位轉變以及凝聚態流速等研究。接著,我們將位能侷限擴展到成為一個一維度的周期性位能,探討其非線性能帶;嘗試了解多能階凝聚態間的可能競爭情形。最後,我們探討自旋凝聚態的同步效應,當激發光束是一個可調極化光的話,如何調控自旋凝聚態的非均勻極化分佈,以及外加的高斯位能對自旋凝聚態的影響。

Exciton-polaritons are half-light and half-matter quasi-bosons, arising in a two-dimensional semiconductor microcavity from the strong-coupling between an exciton and a photon. Exciton-polariton condensate in the microcavities are formed through the stimulated accumulation of low-energy polaritons above a threshold pumping because of the intrinsic light mass from the phtonic component and the interaction from the excitonic component. The features indicating the condensation include the temporal or spatial coherence that extends over th entire condensate, the appearance of quantized vortices, the modified excitation spectrum, as well as the robustness of the condensate profile againist the disorders (superfluidity), and so on. Therefore it is essential to study the dynamics of polariton condensates in order to search for the rich or promising physical effects that could provide further experimental demonstration. This thesis uses the complex Gross-Pitaevskii equation to study the condensate states of the exciton-polaritons with a finite-size pumping spot. Two methods are used to deal with the dynamics of the condensates. One is using a single partial differential equation, incorporating the pumping gain and dissipative terms directly into the conventional Gross-Pitaevskii equation; and the other considers the relaxation mechanism from the reservoir and the polariton is consistently replenished from the reservoir. After the polariton condensate states under a harmonic confinement have been numerically calculated, we studied the dynamic instabilities for a given pumping scheme by using the Bogoliubov perturbation theory. A phase diagram was therefore obtained with respect to the pump power and pump spot. Under this framework, the superfluid properties such as drag force and critical velocities, quantized vortices, BKT transition, and super-currents can be separately discussed. Next, the system is expanded in an one-dimensional periodic potential. We obtain the nonlinear band structures, from which we understand how the transition between 0-state and -state occurs. We also gave the evidences of the synchronization conditions of a two-component spinor condensate with self-interaction and inter-component hopping. Finally, we studied the half-quantum vortices in a pumping scheme with tunable degree of polarization. The inhomogeneous polarization texture and the effects of a Gaussian pinning were studied.

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