玻色-爱因斯坦凝聚体非线性耦合常数在离域阈值处的标度律
Scaling laws for the non-linear coupling constant of a Bose-Einstein condensate at the threshold of delocalization
摘要：
我们探讨了准一维，准二维和三维超冷气体的本地化沿相应的“自由”方向/s的有限范围的缺陷。利用描述玻色-爱因斯坦凝聚体的非线性薛定谔方程计算了最大非线性耦合常数g_max，从而计算了缺陷势能局域化的最大原子数N_max。一个分析模型，基于T-Fermi近似，介绍了波函数。我们表明，g_max成为一个函数的R_0 sqrt（V_0）的各种一维，二维和三维缺陷形状的深度V_0和特征长度R_0。我们的显式计算在很宽的V_0和R_0范围内与这个粗糙模型惊人地一致。一个标度规则也被发现的基态的波函数在阈值处的局域态接近离域。这意味着具有相同乘积R_0 sqrt（V_0）的两个缺陷将因此以相同的g_max彼此相关，并且将在自由方向上具有相同的（减小的）密度分布。
英文摘要：
We explore the localization of a quasi-one-, quasi-two-, and three-dimensional ultra-cold gas by a finite-range defect along the corresponding 'free'-direction/s. The time-independent non-linear Schroedinger equation that describes a Bose-Einstein condensate was used to calculate the maximum non-linear coupling constant, g_max, and thus the maximum number of atoms, N_max, that the defect potential can localize. An analytical model, based on the Thomas-Fermi approximation, is introduced for the wavefunction. We show that g_max becomes a function of R_0 sqrt(V_0) for various one-, two-, and three-dimensional defect shapes with depths V_0 and characteristic lengths R_0. Our explicit calculations show surprising agreement with this crude model over a wide range of V_0 and R_0. A scaling rule is also found for the wavefunction for the ground state at the threshold at which the localized states approach delocalization. The implication is that two defects with the same product R_0 sqrt(V_0) will thus be related to each other with the same g_max and will have the same (reduced) density profile in the free-direction/s.
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