質量重分配及其在自旋-1玻色-愛因斯坦凝聚基態上之應用

Abstract

自旋1玻色愛因斯坦凝聚是一類特殊的，含有三個分量函數的系統。通常以 $Psi=(psi_1,psi_0,psi_{-1})$ 表示。它的行為由一個能量泛函 $E[Psi]$ 及兩個限制條件所描述。這兩個限制分別是原子數守恆與磁化量守恆，也就是說 $int |Psi|^2$ 及 $int (|psi_1|^2-|psi_{-1}|^2)$ 是兩個固定的數。而所謂的基態即指在這兩個條件之下，使能量 $E$ 達到最小的狀態 $Psi$。要解釋這篇論文所討論的問題，首先我們還得指出，根據能量 $E$ 的表達式裡的某個參數的正負號，自旋1玻色愛因斯坦凝聚被分成兩類：順磁性與反磁性。這篇論文裡所討論的工作，其動機來自於下面兩個現象。 1. 任何順磁性系統中的基態，必定滿足下列形式 begin{align*} Psi = (gamma_1 psi,gamma_0 psi,gamma_{-1} psi), end{align*} 其中 $gamma_j$ 皆為常數，而 $psi$ 為函數。這個形式稱作單模近似。 2. 考慮外加一個均勻磁場的情形。若將磁場的強度由零慢慢增加，當強度超過某個特定的數值時，反磁性系統的基態會經歷一個從 $psi_0 equiv 0$ 到 $psi_0 ne 0$ 的分歧。 雖然這兩個現象很早就已經在數值模擬中被發現，但在我們的研究之前，還沒有一個真正嚴格的數學證明。這篇論文包含我們在 [16,17] 這兩篇論文裡的工作，它們分別給出了上面兩個現象的嚴格證明。比起兩篇原本的論文，在本文中我們盡可能把所有的細節都交待清楚。我們的證明方法主要是使用了下面這個原理：質量密度(也就是 $|psi_1|^2$, $|psi_0|^2$ 及 $|psi_{-1}|^2$)的重分配將必定導致動能的下降。這個原理可視為某個廣為人知的梯度的凸性不等式的簡單推廣。我們將會說明這個簡單的原理如何給出解決上面問題的一個統一的想法。

Spin-1 Bose-Einstein condensate (BEC) is a special three-component system, written as $Psi=(psi_1,psi_0,psi_{-1})$. Its behavior is described by an energy functional $E[Psi]$ with two constraints: the conservation of the number of atoms and the conservation of total magnetization. That is $int |Psi|^2$ and $intlt(|psi_1|^2-|psi_{-1}|^2t)$ are fixed numbers. And a ground state is a minimizer of $E$ under the constraints. To explain what we do in this thesis, we remark that according to the sign of a specific parameter in the energy $E$, spin-1 BECs are classified into two groups: ferromagnetic ones and antiferromagnetic ones. The works in this thesis are motivated by the following two phenomena: 1. Any ground state of a ferromagnetic system is of the form begin{align*} Psi = (gamma_1 psi,gamma_0 psi,gamma_{-1} psi), end{align*} where $gamma_j$ are constants and $psi$ a function. This is called single-mode approximation. 2. When an external magnetic field is applied, the ground state of an antiferromagnetic system undergoes a bifurcation from $psi_0 equiv 0$ to $psi_0 ne 0$ as the strength of the magnetic field surpasses a critical value. Although these phenomena have been well-known from numerical simulations for quite a long time, there were no rigorous mathematical justifications before our investigations. In this thesis, our works [16,17] on their proofs are given, with more details. The proofs rely on a principle which says that a redistribution of the mass densities (i.e. $|psi_1|^2$, $|psi_0|^2$ and $|psi_{-1}|^2$) will decrease the kinetic energy. This principle can be regarded as a simple generalization of a well-known convexity inequality for gradients. We will show how this principle can give a rather unified approach toward our problems.