非旋转冷原子玻色-爱因斯坦凝聚体中的Poincaré指数公式及其与Kosterlitz-Einstein无相变的类比
Poincaré index formula and analogy with the Kosterlitz-Thouless transition in a non-rotated cold atom Bose-Einstein condensate
摘要：
本文用含时Gross-Pitaevskii方程模拟了非旋转轴对称简谐势阱中玻色-爱因斯坦凝聚原子的稀薄气体。当凝聚体携带的角动量不为零时，最小能态描述了围绕陷阱中心传播的涡旋（或反涡旋）。（反）涡旋的数量随着角动量的增加而增加，它们相互排斥形成Abrikosov晶格。除了涡和反涡之外，还有滞流点，在那里超流消失;据我们所知，滞流点以前没有在Gross-Pitaevskii方程中分析过。庞加莱指数公式表明，涡和驻点的数量之差永远不会改变。当驻点数目较少时，它们倾向于聚集成简并传播结构。但是当这个数目变得足够大时，驻点倾向于与涡核配对，以规则的晶格排列围绕陷阱中心传播。这与Kosterlitz-无相变的几何形状类似，凝结体的角动量代替温度作为外部控制参数。
英文摘要：
A dilute gas of Bose-Einstein condensed atoms in a non-rotated and axially symmetric harmonic trap is modelled by the time dependent Gross-Pitaevskii equation. When the angular momentum carried by the condensate does not vanish, the minimum energy state describes vortices (or antivortices) that propagate around the trap center. The number of (anti)vortices increases with the angular momentum, and they repel each other to form Abrikosov lattices. Besides vortices and antivortices there are also stagnation points where the superflow vanishes; to our knowledge the stagnation points have not been analyzed previously, in the context of the Gross-Pitaevskii equation. The Poincaré index formula states that the difference in the number of vortices and stagnation points can never change. When the number of stagnation points is small, they tend to aggregate into degenerate propagating structures. But when the number becomes sufficiently large, the stagnation points tend to pair up with the vortex cores, to propagate around the trap center in regular lattice arrangements. There is an analogy with the geometry of the Kosterlitz-Thouless transition, with the angular momentum of the condensate as the external control parameter instead of the temperature.
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