Josephson Dynamics of 2D Bose-Einstein Condensates in Dual-Core Trap: Homogeneous, Droplet-Droplet, and Vortex-Vortex Regimes

TL;DR

The paper addresses Josephson dynamics in a two-dimensional, two-component Bose-Einstein condensate loaded into a symmetric dual-core trap, including beyond-mean-field Lee-Huang-Yang corrections. It combines an extended GPE model with a dimer reduction for the homogeneous case, a variational approach for quantum droplets, and direct simulations to map the nonlinear dynamics. Key contributions include analytical predictions for zero- and π-phase Josephson frequencies, a bifurcation structure featuring two pitchfork points in the zero-phase branch and one in the π-phase, and demonstration of Andreev-Bashkin-type drag in both droplets and vortices along with stability analyses. The results establish a comprehensive framework for observing LHY-stabilized Josephson phenomena in quasi-2D dual-core BECs, with experimentally accessible parameter estimates and clear implications for droplet and vortex dynamics.

Abstract

The dynamics of a two-dimensional Bose-Einstein condensate mixture, loaded into a dual-core trap, when beyond-mean-field effects are taken into account, are considered. The effects of quantum fluctuations are described by the Lee-Huang-Yang correction terms in the extended coupled Gross-Pitaevskii equations. The spatially uniform and inhomogeneous BEC cases are studied. In the first case, the parameter regimes associated with macroscopic quantum tunnelling, self-trapping, and revival-like localisation dynamics are found. The Josephson oscillation frequencies for both the zero-phase and the $π$-phase modes are derived. As the total atom number varies, the dynamics exhibit a nontrivial bifurcation structure: along the zero-phase branch, two pitchfork bifurcations generate bistability and hysteresis, while the $π$-phase branch shows a single pitchfork bifurcation. In the second case, the Josephson dynamics for quantum droplets and vortices are investigated. Predictions for the oscillation frequencies of the atomic population between quantum droplets are obtained and fully validated by direct numerical simulations of coupled extended GP equations. The existence of the Andreev-Bashkin nondissipative drag through simulations of droplet-droplet interactions is shown. The Josephson dynamics of vortex states are studied. Vortices with topological charge $S$ and sufficiently small particle number are typically unstable, breaking up into $S+1$ (occasionally $S+2$) fundamental fragments, with the breakup time increasing as the particle number grows. Unstable asymmetric vortices show splitting and/or crescent-like instability. For vortices with sufficiently large norms, long-time simulations confirm robust stability against small perturbations; in this regime, Josephson oscillations and Andreev-Bashkin-type entrainment for vortex states with charges $S=1, 2$, and $3$ are investigated.

Figures (19)

• Figure 1: Phase portraits in the $(Z,\theta)$ phase plane are shown for different atom numbers $N$ and phase modes. In each panel, the dotted curve corresponds to bounded trajectories (finite motion) representing Josephson oscillations, the separatrix is traced by the critical initial imbalance $Z_0=Z_{0\mathrm{cr}}$ (solid line), and the dashed curve represents unbounded trajectories (infinite motion) associated with the self-trapping regime. (a) Zero-phase mode, $N=1.2$: $Z_0=0.3$ (dotted, Josephson oscillations), $Z_0=Z_{0\mathrm{cr}}\simeq 0.6251$ (solid, separatrix), and $Z_0=0.75$ (dashed, self-trapping). (b) $\pi$-phase mode, $N=1$: $Z_0=0.55$ (dotted, Josephson oscillations), $Z_0=Z_{0\mathrm{cr}}\simeq 0.999$ (solid, separatrix), and $Z_0=0.9898$ (dashed, self-trapping). (c) $\pi$-phase mode, $N=0.8$: $Z_0=0.5$ (dotted, Josephson oscillations), $Z_0=Z_{0\mathrm{cr}}\simeq 0.9517$ (dashed, separatrix), and $Z_0=0.995$ (solid, self-trapping). Other parameters $q=1$, $g=1$ and $\kappa=0.01$.
• Figure 2: The time evolution of the population imbalance $Z(t)$ is shown for three representative initial conditions: (a) $Z_0=0.3$, which yields bounded motion corresponding to Josephson oscillations; (b) $Z_{0\mathrm{cr}}=0.6251$, which produces the critical trajectory on the separatrix; and (c) $Z_0=0.75$, which results in unbounded (running-phase) dynamics associated with the self-trapping regime. The solid and dashed curves represent the theoretical predictions and the numerical simulations, respectively. Panel (d) presents the corresponding phase dynamics $\theta(t)$ for the same initial imbalances. The Josephson-oscillation regime is identified by small-amplitude oscillations (blue dashed curve), while larger-amplitude oscillations occur on the separatrix (blue solid curve). In the self-trapping regime, $\theta(t)$ exhibits running-phase behaviour (red solid curve; note the right vertical axis). These time evolutions correspond to the trajectories shown in the $(Z,\theta)$ phase portrait in Fig. \ref{['fig:phaseport']}(a).
• Figure 3: (a) The time evolution of the relative phase difference $\theta(t)$ (top panels) and the population imbalance $Z(t)$ (bottom panels) in the localization-revival regime is shown. Solid curves represent the theoretical predictions, while dashed curves correspond to direct numerical simulations. (b) The corresponding dynamics of the atom numbers in each component, $N_1(t)$ (lower curve) and $N_2(t)$ (upper curve), is displayed. The parameter set is the same as for the dashed trajectory in the phase portrait of Fig. \ref{['fig:phaseport']}(c).
• Figure 4: The dependence of the small-amplitude Josephson oscillation frequency on the total number of atoms $N$ is shown for (a) the zero-phase mode and (b) the $\pi$-phase mode. The solid curves correspond to the analytical predictions given by Eqs. (\ref{['eq:Jfzero']}) and (\ref{['eq:Jfpi']}), while the points represent results obtained from direct numerical simulations of Eq. (\ref{['eq:gpe']}). The remaining parameters are fixed, with $q=1$, $g=1$, and $\kappa=0.01$.
• Figure 5: Bifurcation diagram in the $(N, Z)$-plane. (a) In the zero-phase configuration $(\sigma = 1)$, the first bifurcation is supercritical and occurs at $N_{b,1} \simeq 0.0239252$, as determined by Eq. (\ref{['eq:Nb']}). (b) As $N$ increases further in the zero-phase, a second bifurcation of subcritical type emerges at $N_{b,2} \simeq 0.847427$. Additionally, a saddle-node bifurcation is identified at $N = N_s \simeq 0.876157$, giving rise to a hysteresis loop within the interval $N_{b,2} < N < N_s$. (c) In the $\pi$-phase $(\sigma = -1)$, a single supercritical bifurcation takes place at $N_b \simeq 0.927631$, again as predicted by Eq. (\ref{['eq:Nb']}). The remaining parameters are fixed as $\kappa = 0.01$, $q = 1$, and $g = 1$.