We revisit the problem of the reduction of the three-dimensional (3D) dynamics of Bose-Einstein condensates, under the action of strong confinement in one direction ($z$), to a 2D mean-field equation. We address this problem for the confining potential with a singular term, viz ., $V_z(z)=2z^2+{\zeta }^2/z^2$, with constant $$. A quantum phase transition is induced by the latter term, between the ground state (GS) of the harmonic oscillator and the 3D condensate split in two parallel non-interacting layers, which is a manifestation of the superselectioneffect. A realization of the respective physical setting is proposed, making use of resonant coupling to an optical field, with the resonance detuning modulated along $z$. The reduction of the full 3D Gross-Pitaevskii equation (GPE) to the 2D nonpolynomial Schr"{o}dinger equation (NPSE) is based on the factorized ansatz, with the $z$ -dependent multiplier represented by an exact GS solution of the 1D Schr"{o}dinger equation with potential $V(z)$. For both repulsive and attractive signs of the nonlinearity, the 2D NPSE produces GS and vortex states, that are virtually indistinguishable from the respective numerical solutions provided by full 3D GPE. In the case of the self-attraction, the threshold for the onset of the collapse, predicted by the 2D NPSE, is also virtually identical to its counterpart obtained from the 3D equation. In the same case, stability and instability of vortices with topological charge $S=1$, $ 2 $, and $3$ are considered in detail. Thus, the procedure of the spatial-dimension reduction, 3D $$ 2D, produces very accurate results, and it may be used in other settings.

中文翻译：

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双层Bose-Einstein冷凝物：横向上的量子相变并还原为二维

我们重新讨论了在一个方向上的强约束作用下玻色-爱因斯坦凝聚物三维（3D）动力学降低的问题（$ž$）转换为2D平均场方程。我们用单数项（即，$V_{ž}（ž）=2{ž}^2+{\zeta }^2/{ž}^2$，并具有恒定的$$。后一项在谐波振荡器的基态（GS）和3D冷凝物分成两个平行的非相互作用层之间引起量子相变，这是超选择效应的体现。提出了一种相应的物理设置的实现，它利用与光场的共振耦合，其中共振失谐沿着$ž$。将完整的3D Gross-Pitaevskii方程（GPE）简化为2D非多项式Schr“ {o} dinger方程（NPSE）是基于分解的ansatz，其中$ž$ 一维Schr“ {o} dinger方程的精确GS解表示的势相关乘子 $V（ž）$。对于非线性的排斥和吸引人的征兆，2D NPSE会产生GS和涡旋状态，这与全3D GPE提供的相应数值解几乎没有区别。在自我吸引的情况下，由2D NPSE预测的崩溃开始的阈值实际上也与从3D方程获得的阈值相同。在相同情况下，带有拓扑电荷的涡旋的稳定性和不稳定性$\mathrm{小号}=\mathrm{1个}$，$ 2 $和 $3$被详细考虑。因此，空间维数减少过程3D $$ 2D产生非常准确的结果，并且可以在其他设置中使用。