快速旋转的玻色-爱因斯坦凝聚体向巨涡相的转变
The Transition to a Giant Vortex Phase in a Fast Rotating Bose-Einstein Condensate
摘要：
研究了二维单位圆盘上快速旋转玻色-爱因斯坦凝聚体的Gross-Pitaevskii（GP）能量泛函.将耦合参数写为$ 1 / \eps^2 $，我们考虑渐近状态$ \eps \to 0 $，其中角速度$Ω$与$ (\eps^2|\log\eps|)^{-1} $成比例。我们证明，如果$ Ω= Ω_0 (\eps^2|\log\eps|)^{-1} $和$ Ω_0 > 2(3π)^{-1} $，那么最小的GP能量泛函没有零的环在边界处的光盘，包含大部分的质量。涡量存在于中心周围的一个互补的“洞”中，在那里密度小得几乎为零。此外，我们证明了基态能量的下界匹配，直到小的误差，从最佳巨涡试探函数得到的上界，也是一个GP极小盘周围的缠绕数是符合这个试探函数的相位。
英文摘要：
We study the Gross-Pitaevskii (GP) energy functional for a fast rotating Bose-Einstein condensate on the unit disc in two dimensions. Writing the coupling parameter as $ 1 / \eps^2 $ we consider the asymptotic regime $ \eps \to 0 $ with the angular velocity $Ω$ proportional to $ (\eps^2|\log\eps|)^{-1} $. We prove that if $ Ω= Ω_0 (\eps^2|\log\eps|)^{-1} $ and $ Ω_0 > 2(3π)^{-1} $ then a minimizer of the GP energy functional has no zeros in an annulus at the boundary of the disc that contains the bulk of the mass. The vorticity resides in a complementary `hole' around the center where the density is vanishingly small. Moreover, we prove a lower bound to the ground state energy that matches, up to small errors, the upper bound obtained from an optimal giant vortex trial function, and also that the winding number of a GP minimizer around the disc is in accord with the phase of this trial function.
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