渦旋在type-Ⅱ超導體和旋轉超流體中都是重要現象。在type-Ⅱ超導體中，當磁場處於第一和第二臨界場之間時 (H_c1≤H≤H_c2)，可以觀察到渦旋。但是當H_c2不夠高或實驗磁場不容易接近H_c2時，觀察到渦旋晶格的形成可能會有困難。在旋轉超流體中，例如簡諧位能中的冷原子，也有類似的問題。當旋轉頻率Ω不容易接近諧振頻率ω時，在旋轉超流體中觀察到渦旋晶格的形成也可能有困難。[1]史丹佛大學Alenxendar Fetter教授在2001年提出在旋轉超流體中除了均向性的簡諧位能外再加入一個均向四次位能井可以克服這可能的問題。在這篇論文裡，我們研究超快速旋轉的玻色-愛因斯坦凝聚物中的渦旋結構，我們考慮該凝聚物處在一非均向性的簡諧位能加上一均向性的四次位能井。我們進一步考慮該凝聚物在z方向上受到高度限制，並且旋轉是繞著z軸進行，所以簡諧位能的非均向性是表現在x-y平面。這種非均向性可由比值γ ≡ ωy/ωx (其中ωx和ωy分別表示x和y方向的諧振頻率）來描述。在超快速旋轉的條件下Ω > ωx (其中Ω是旋轉頻率），且當四次位能井強度λ很小時，我們觀察到當γ很小時，橢圓環中會形成環狀渦旋。隨著γ的增加，渦旋可以展現成對結構，進而形成三角晶格。而當λ較大時，不論γ的值如何，渦旋將以橢圓結構形成，呈現出三角晶格。除了數值計算外，我們也應用托馬斯-費米近似作解析的探討。研究發現，托馬斯-費米近似可以給出這個系統的渦旋結構的良好描述。

Vortices play a crucial role in both type-II superconductors and rotating superfluids. In type-II superconductors, vortices can be observed when the magnetic field is lying between the first and second critical fields (Hc_1≤H≤Hc_2). However, when Hc_2 is too low or when Hc_2 is too high to access, observing the vortex lattice can be challenging. Similar difficulties can arise in rotating superfluids, such as the cold atoms trapped in a harmonic potential when the rotational frequency Ω is not easily approaching the trap frequency. In 2001, Professor Alexander Fetter [1] of Stanford University proposed that adding a quartic potential to the isotropic harmonic trap in rotating superfluids can overcome this problem. In this thesis, we study the vortex structures in a rapidly rotating condensate in an anisotropic harmonic plus isotropic quartic trap. The condensate is considered to be highly confined in the z-direction, and the rotation is around the z-axis. Thus, the harmonic potential anisotropy is characterized in the x-y plane via the aspect ratio γ ≡ ωy/ωx ≥ 1 (ℏω_x is the energy scale in our study). Under ultra-rapid rotation, Ω > ωx and when the quartic trap strength λ is small, annular vortices will form in an oval ring when γ is small. As γ increases, vortices can exhibit pairwise structures that a triangular lattice can develop. When λ is larger, regardless of the value of γ, vortices will form in an elliptical structure exhibiting a triangular lattice. In addition to numerical calculations, we perform analytic works and show that the Thomas-Fermi approximation can give a good description of the vortex structures upon changing the parameters.

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