Beyond mean-field effects in Josephson oscillations and self-trapping of Bose-Einstein condensates in two-dimensional dual-core traps

TL;DR

The paper addresses beyond-mean-field effects in Josephson dynamics of binary Bose-Einstein condensates confined in a two-dimensional dual-core trap, incorporating Lee-Huang-Yang corrections. It analyzes both spatially uniform condensates and quantum droplets, deriving analytical Josephson frequencies for zero- and π-phase modes and mapping the resulting bifurcation structures, including hysteresis in the zero-phase and a single bifurcation in the π-phase. For droplets, a variational approach yields stationary parameters and Josephson frequencies, with numerical simulations validating the results and revealing limitations at large norms. The study also investigates moving droplets and inter-droplet interactions, identifying phase-dependent forces and demonstrating Andreev-Bashkin drag in the two-core geometry. Overall, the work clarifies how beyond-mean-field fluctuations reshape macroscopic quantum tunneling, self-trapping, and droplet dynamics in low-dimensional settings with potential experimental relevance.

Abstract

We study a binary Bose gas in a symmetric dual-core, pancake-shaped trap, modelled by two linearly coupled two-dimensional Gross-Pitaevskii equations with Lee-Huang-Yang corrections. Two different cases are considered. First, we consider a spatially uniform condensate, where we identify the domains of parameters for macroscopic quantum tunnelling, self-trapping and localisation revivals. The analytical formulas for the Josephson frequencies in the zero- and $π$-phase modes are derived. As the total atom number varies, the system displays a rich bifurcation structure. In the zero-phase, two successive pitchfork bifurcations generate bistability and hysteresis, while the $π$-phase exhibits a single pitchfork bifurcation. The second case is when the quantum droplets are in a dual-core trap. Analytical predictions for the oscillation frequencies are derived via a variational approach for the coupled dynamics of quantum droplets, and direct numerical simulations validate the results. We identify critical values of the linear coupling that separate Josephson and self-trapped regimes as the particle number changes. We also found the Andreev-Bashkin superfluid drag effect in numerical simulations of the droplet-droplet interactions in the two-core geometry.

Figures (15)

• Figure 1: These phase portraits are presented for different initial imbalances and different phase modes: $Z_0=0.04$ (dotted line), $0.0553$ (corresponds to a solid line and a critical value), $0.065$ (dashed line), $\theta_0=0$, $N=10$ (a), $Z_0=0.3$ (dotted line), $0.6829$ (corresponds to a solid line and a critical value), $0.8$ (dashed line), $\theta_0=\pi$, $N=0.6$ (b), $Z_0=0.5$ (dotted line), $0.99$ (solid line), $0.004$ (dashed line), $\theta_0=\pi$, $N=0.85$ (c). Other parameters $K=0.01$ and $g=1$.
• Figure 2: The dynamics of the atomic imbalance $Z(t)$ for different initial values: $Z_0=0.04$ (a), $Z_{0c}=0.0553$ (b), and $Z_0=0.065$ (c). The solid and dashed lines correspond to the numerical simulations and the theoretical results, respectively. The phase dynamics corresponding to these initial values of the atomic imbalance are shown in panel (d). The dashed line ($Z_0=0.04$) and the dotted line ($Z_0=0.0553$) correspond to the left vertical axis, while the solid line ($Z_0=0.065$) corresponds to the right vertical axis. For all panels: $\theta_0=0$, $N=10$$K=0.01$ and $g=1$.
• Figure 3: (a) The dynamics of $Z(t)$ (bottom plots) and relative phase $\theta(t)$ (upper plots) in the localisation revival mode are presented. The solid and dashed lines show the theoretical predictions and the numerical results, respectively. (b) Time evolution of the number of particles in each component. The bottom curve for $N_1$ and the upper curve for $N_2$. The parameters coincide with those of the dashed trajectory in the phase portrait of Fig. \ref{['fig1']}(c).
• Figure 4: The dependence of the Josephson frequency on the number of atoms is presented for zero-(a) and $\pi$-phase (b) modes. The solid lines and the points correspond to the Eqs. (\ref{['eq:Jfzero']}, \ref{['eq:Jfpi']}) and numerical simulation results, respectively. Other parameters $K=0.01$ and $g=1$.
• Figure 5: Bifurcation diagram in the $(N, Z)$-plane. (a) In the zero-phase, the first supercritical bifurcation occurs at $N_{b,1} \simeq 0.0038$, corresponding to $\sigma = 1$ in Eq. (\ref{['eq:Ncr']}) and associated with the lower branch of the Lambert function $W_{-1}$. (b) As $N$ increases further in the zero-phase, the second subcritical bifurcation occurs at $N_{b,2} \simeq 0.7155$, determined by the principal branch $W_0$ of the Lambert function. A saddle-node bifurcation appears at $N = N_s \simeq 0.8665$, and hysteresis is observed in the interval $N_{b,2} < N < N_s$. (c) In the $\pi$-phase, a subcritical bifurcation occurs at $N_b \simeq 0.7555$, corresponding to $\sigma = -1$ and obtained via the principal branch $W_0$ in Eq. (\ref{['eq:Ncr']}). The other parameters are fixed as $K = 0.01$ and $g = 1$.