Dos Santos Mateus C P, Malomed Boris A, Cardoso Wesley B
Instituto de Física, Universidade Federal de Goiás 74.690-970, Goiânia, Goiás, Brazil.
Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, and Center for Light-Matter Interaction, Tel Aviv University, Tel Aviv 69978, Israel.
Phys Rev E. 2020 Oct;102(4-1):042209. doi: 10.1103/PhysRevE.102.042209.
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}+ζ^{2}/z^{2}, with constant ζ. 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 noninteracting layers, which is a manifestation of the "superselection" effect. 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 Schrödinger equation (NPSE) is based on the factorized ansatz, with the z -dependent multiplier represented by an exact GS solution of the 1D Schrödinger equation with potential V_{z}(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.
我们重新审视了玻色 - 爱因斯坦凝聚体三维（3D）动力学在一个方向（z）的强约束作用下简化为二维平均场方程的问题。我们针对具有奇异项的约束势来解决这个问题，即(V_{z}(z)=2z^{2}+ζ^{2}/z^{2})，其中(ζ)为常数。后一项会在简谐振子的基态（GS）与分裂为两个平行非相互作用层的三维凝聚体之间引发量子相变，这是“超选择”效应的一种表现。我们提出了一种利用与光场的共振耦合来实现相应物理设置的方法，其中共振失谐沿z方向调制。将完整的三维格罗斯 - 皮塔耶夫斯基方程（GPE）简化为二维非多项式薛定谔方程（NPSE）是基于因式分解假设，其中与z相关的乘数由具有势(V_{z}(z))的一维薛定谔方程的精确GS解表示。对于非线性的排斥和吸引符号，二维NPSE都会产生GS态和涡旋态，这些态与完整三维GPE提供的相应数值解几乎无法区分。在自吸引的情况下，二维NPSE预测的坍缩起始阈值也与从三维方程得到的阈值几乎相同。在相同情况下，我们详细考虑了拓扑电荷(S = 1)、(2)和(3)的涡旋的稳定性和不稳定性。因此，从三维到二维的空间维度简化过程产生了非常准确的结果，并且它可以用于其他设置。
