Josephson Oscillation and Nonlinear Self-Trapping in Quasi-one-dimensional Quantum Liquid

TL;DR

The work analyzes Josephson oscillation and self-trapping in a binary Bose-Einstein condensate confined to quasi-one dimension, using a double-well potential and a two-mode reduction that includes mean-field ($g_3$), beyond mean-field ($g_4$), and three-body ($g_5$) interactions. It combines a two-mode dynamical framework with Bogoliubov theory to map JO/ST behavior, phase-space structures, and excitation spectra in both symmetric and asymmetric wells across Q1D and 1D reductions, highlighting instabilities and roton-like dispersions. The main findings show that attractive mean-field promotes self-trapping while repulsive beyond mean-field and three-body terms oppose it, with the Josephson frequency depending nonlinearly on particle number $N$ and interaction strengths; small tilt enhances the visibility of interaction effects and improves experimental accessibility. The Bogoliubov spectra reveal roton-like kinks at low density, suggesting precursors to supersolid-like phases, and the results provide quantitative predictions for JO frequency and critical imbalance across dimensionalities and interaction regimes, offering guidance for future experiments on quantum liquids in reduced dimensions.

Abstract

In this article, we study the two-mode method to analyze the Josephson oscillation for a trapped binary Bose-Einstein condensate while taking into account the beyond mean-field and three body interactions. For this purpose, we use the archetypal model of double well potential and study the Josephson oscillation and self-trapping phases in quasi-one dimension. Additionally, our analysis provides quantitative discussion on the effect of asymmetry and dimension. We further corroborate our findings with Bogoliubov quasi-particle method and notice regions of instabilities and roton like mode.

Figures (10)

• Figure 1: Schematic representation of the physical situation for particles trapped in a double well potential. The blue circles describes the atoms. $U_{eff}$ includes MF, BMF and 3B interactions. $V_0$ is the barrier height.
• Figure 2: (color online) Representation of condensate wave function $\psi$ of the system in a double well potential by considering all the mentioned interactions. The coupling strengths were considered as unity in magnitude albeit the nature being different. The MF and BMF interactions are attractive and repulsive respectively. The 3B interaction is also taken as repulsive.
• Figure 3: (color online) Figure describes the phase portrait for Q1D system. Solid blue curve notes the presence of MF interaction only for initial critical imbalance $z_{0} = 0.038$. Red dashed curve takes into account MF and BMF for $z_{0} =0.0484$, while green dotted curve depicts all interaction combination i.e. MF, BMF and 3B when we fix the initial imbalance, $z_{0}$ at $0.089$. The purple dashed-dotted curve shows the combination of MF and 3B interactions for $z_{0} = 0.0754$.
• Figure 4: (color online) Description of various combination of interaction strength in Q1D system, which describe how the system reacts from Josephson to self trapping regime. In (a) and (b) only MF is considered while in (c), (d) combination of MF and BMF correction is taken into account. (e) and (f) is prepared when all the interaction term MF + BMF + 3B are present and (g), (h) describes only MF and 3B interaction. Here, $g_3=-0.6$, $g_4=0.01$ and $g_5=0.009$. All these values are in arbitrary units. $N$ is taken as $11.5$.
• Figure 5: (color online) Description of various combination of interaction strength in 1D system, which describe how the system reacts from Josephson to self trapping regime. In (a) and (b) only MF is considered while in (c), (d) combination of MF and BMF correction is taken into account. (e) and (f) is prepared when all the interaction term like MF, BMF and 3B are present and (g), (h) describes only MF and 3B interaction. Here, $g_3=0.6$, $g_2=-0.01$ and $g_5=0.009$. All these values are in arbitrary units. $N$ is taken as $11.5$.