Faraday waves on a bubble Bose-Einstein condensed binary mixture

TL;DR

This work investigates dynamic stability and pattern formation for a binary BEC confined to a spherical bubble, where a time-periodic Rabi coupling drives parametric resonances on a discrete angular spectrum. A reduced 2D spherical GP model yields exact analytical population dynamics via a Duffing-type equation for the density difference, complemented by BdG and Floquet stability analyses and full GP simulations. The key finding is that Faraday waves can emerge and coexist with an immiscible phase, with Floquet theory revealing instabilities and resonance conditions that BdG misses at higher drive strengths. The results offer a mechanism to control spatial patterns in closed 2D geometries and suggest experimental routes in ultracold bubble-like systems, highlighting the pivotal role of $s$-wave interactions and Rabi coupling in driving these phenomena.

Abstract

By studying the dynamic stability of Bose-Einstein condensed binary mixtures trapped on the surface of an ideal two-dimensional spherical bubble, we show how the Rabi coupling between the species can modulate the interactions leading to parametric resonances. In this spherical geometry, the discrete unstable angular modes drive both phase separations and spatial patterns, with Faraday waves emerging and coexisting with an immiscible phase. Noticeable is the fact that, in the context of discrete kinetic energy spectrum, the only parameters to drive the emergence of Faraday waves are the $s-wave$ contact interactions and the Rabi coupling. Once analytical solutions for population dynamics are obtained, the stability of homogeneous miscible species is investigated through Bogoliubov-de Gennes and Floquet methods, with predictions being analysed by full numerical solutions applied to the corresponding time-dependent coupled formalism.

Figures (15)

• Figure 1: (Color online) In panel (a), the density oscillating period $T$ is given as a function of the absolute difference of the interaction parameters $|\Delta g|$, for three different Rabi couplings $\Omega$, as indicated. In panel (b), it is shown the perfect agreement between analytical expressions for the Duffing period $T_K$ (empty-triangles) and $T$ (solid-line), respectively multiplied by $\Omega$, given by \ref{['period-ell']} and \ref{['period']}, with $\alpha=\frac{{\cal A}\Delta g}{8\pi}$. Within defined units, all quantities are presented as dimensionless.
• Figure 2: (Color online) Time-evolution of the atom-number ratio, $N_j(t)/N\equiv 4\pi|\psi_{j}(t)|^{2}\equiv 4\pi n_j$, with initial condition $N_j(0)=N/2$, for given Rabi couplings $\Omega$. In both panels, (a) for $|\Delta g|=0$, and (b) for $|\Delta g|=10$, the initially decreasing (increasing) lines refer to species 1 (species 2) [horizontal line for $\Omega=0$]. In (b), the full-numerical solutions (legend box), are matching with corresponding solid lines, given by \ref{['atom-diff']}, for $({\cal A},\Omega) =(0.7873,\;0.2)$, $(0.6261,\;0.1)$, and $(0.2214,\;0.01)$, with $(\nu_A/{\cal A})=\pi/\left(\sqrt{2} K(-1)\right)=1.6945$. Within defined units, all quantities are presented as dimensionless.
• Figure 3: (Color online) BdG stability diagrams for the interaction parameters $g_{12}$ vs $g$, as given by \ref{['bdg_spc']}. Stable regions [Im($\omega_{\ell,\pm}$) = 0] are represented in black, with unstable $\ell-$modes [Im$(\omega_{\ell,\pm})$$\neq0$] (with $\ell$ values indicated inside the regions) are in colors [violet for $\ell=1$ and orange for $\ell=2$]. The Rabi coupling $\Omega=0.1, 0.9$ are indicated at the top of the respective panels. Within defined units, all quantities are presented as dimensionless.
• Figure 4: (Color online) Floquet stability diagrams for constant couplings, $\Omega=0.1$ (a) and $\Omega=0.9$ (b), parametrized by the interactions $g_{12}$ vs $g$, determined by $(\lambda_\ell^R)_{\rm max}$ [See \ref{['floquet_eq']}-\ref{['floquet_exp']}]. The stable regions [$(\lambda_\ell^R)_{\rm max}\le 0$] are in black, with unstable ones [$(\lambda_\ell^R)_{\rm max}> 0$] having the $\ell-$mode given in colors [violet ($\ell=1$), orange ($\ell=2$) and yellow ($\ell=3$)]. The dashed lines, at $g=20$ (a) and $g=40$ (b) refer to results presented, respectively, in (a) and (b) of Fig. \ref{['fig05']}. Within defined units, all quantities are presented as dimensionless.
• Figure 5: (Color online) Floquet spectra, given by $(\lambda_{\ell}^R)_{\rm max}$, from \ref{['floquet_exp']}, are shown as functions of the inter-species interaction $g_{12}$, with the intra-species and Rabi coupling parameters fixed at ($\Omega$, $g$)$=$ (0.1, 20) (a) and (0.9, 40) (b). The $\ell=$1 and 2 unstable modes are, respectively, represented by solid-violet and dashed-orange lines. In the insets,$(\lambda_{\ell}^R)_{\rm max}$ is re-scaled to improve visibitily of lower peak instability regions. Within defined units, all quantities are presented as dimensionless.