Double-layer Bose-Einstein condensates: A quantum phase transition in the transverse direction, and reduction to two dimensions

TL;DR

This paper investigates the reduction of 3D Bose-Einstein condensate dynamics to a 2D model under strong confinement, revealing a quantum phase transition and validating the 2D approximation's accuracy for various states.

Contribution

It introduces a new reduction method for 3D to 2D BEC equations involving a singular potential, capturing quantum phase transitions and vortex states with high accuracy.

Findings

The 2D NPSE accurately reproduces 3D GPE ground and vortex states.

The collapse threshold in attractive BECs is consistent between 2D and 3D models.

A quantum phase transition occurs between a single ground state and a split two-layer condensate.

Abstract

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 $\zeta$. 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 Schr\"odinger equation (NPSE) is based on the factorized ansatz, with the $z$-dependent multiplier represented by an exact GS solution of the Schr\"odinger 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.