论文标题
双层Bose-Einstein冷凝水：横向方向上的量子相变，还原至二维
Double-layer Bose-Einstein condensates: A quantum phase transition in the transverse direction, and reduction to two dimensions
论文作者
论文摘要
我们重新审视了将玻色子凝结物减少的三维（3D）动力学的问题，在一个方向上强限制（$ z $）的作用下，将其减少到2D平均场方程。我们以单数项为限制电势，即，$ v_ {z}（z）= 2z^{2}+ζ^{2}/z^{2} $，带有常数$ζ$。量子相跃迁是由后一个项诱导的，在谐波振荡器的基态（GS）和在两个平行的非相互作用层中分裂的3D冷凝物之间，这是“超选择”效应的表现。提出了相应的物理设置的实现，利用谐振耦合到光场，并沿$ z $进行了共振引起的调节调节。将完整的3D Gross-Pitaevskii方程（GPE）降低到2D非多物质schrödinger方程（NPSE）是基于分解的ANSATZ，其$ Z $依赖性乘数由schrödinger方程的精确GS溶液代表，具有$ v（z）$ v（z）$。对于非线性的排斥性和有吸引力的迹象，NPSE会产生GS和涡旋状态，这些状态与完整3D GPE提供的各自的数值解决方案几乎没有区别。在自我吸引力的情况下，由2d NPSE预测的崩溃开始的阈值也与从3D方程中获得的对应物几乎相同。在同一情况下，详细考虑了具有拓扑费$ s = 1 $，$ 2 $和$ 3 $的涡流的稳定性和不稳定。因此，空间维度减少的过程（3D $ \ rightarrow $ 2D）产生非常准确的结果，并且可以在其他设置中使用。
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 non-interacting 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 Schrödinger equation with potential $V(z)$. For both repulsive and attractive signs of the nonlinearity, the 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 $\rightarrow$ 2D, produces very accurate results, and it may be used in other settings.
