Bose-Einstein droplet in free space

A “gaseous droplet”—a stable gas that coheres by itself in free space—is shown to be produced with a Bose-Einstein condensate. Collapse and expansion of the droplet are suppressed by rapid oscillations of the interatomic interaction between attractive and repulsive through, e.g., the Feshbach resonance. Energy dissipation, which is present in realistic situations, is found to play a crucial role in suppressing dynamical instabilities that destabilize the droplet.

Figure 1

(a) Time evolution of the peak density ${\mid \psi (r=0)\mid }^2$ (left axis) and monopole moment $⟨r⟩=\int d\mathbf{r}r{\mid \psi \mid }^2$ (right axis) for the oscillating interaction $g\left(t\right)=-69+155\sin t$ with $\gamma =0.3$. The initial state is the noninteracting ground state in a trapping potential $r^2∕1800$. The interaction is gradually turned on, and the trapping potential is simultaneously turned off as shown in the inset. (b) A magnification of (a). The inset shows the density profile ${\mid \psi \left(r\right)\mid }^2$ from $t=8000$ to $t=8020$. The dotted curve represents the oscillatory (sine) part of the interaction in Eq. (4).

Figure 2

Time evolution of the peak density ${\mid \psi (r=0)\mid }^2$ (left axis) and monopole moment $⟨r⟩=\int d\mathbf{r}r{\mid \psi \mid }^2$ (right axis). The initial state is prepared as in Fig. 1a, and $(g_0,g_1)$ are linearly changed from ($-69$, 155) to ($-69$, 160) in (a), ($-69$, 129) in (b), and ($-72$, 173) in (c) during $0⩽t⩽600$. The insets in (b) and (c) are magnifications of ${\mid \psi (r=0)\mid }^2$ in the dashed squares and the density profiles ${\mid \psi \left(r\right)\mid }^2$ at the times indicated by $A$ and $B$.

Figure 3

The lifetime of a BEC droplet after a stable state is prepared as in Fig. 1, and the interaction is linearly changed during $0⩽t⩽600$ as in Fig. 2. (a) The black region corresponds to the one in which a droplet expands during the change of the interaction, and the white region corresponds to the one in which a droplet survives until at least $t=10600$. The behaviors in the regions labeled as “instability I, II, and III” correspond to those in Figs. 2a, 2b, 2c, respectively. (b) The parameter $\gamma$ is also linearly changed from $\gamma =0.03$ to 0.01 during $0⩽t⩽600$ as well as the interaction.

Figure 4

The profiles of the column density of a BEC droplet obtained by full 3D numerical calculations. Since the isotropic symmetry is preserved, we obtain the same column density from any direction. The size of the images is $12\times 12$ in units of ${(\hslash ∕m\Omega )}^{1∕2}$.

Figure 5

The density profiles ${\mid \psi \left(r\right)\mid }^2$ (solid curves) of the stable droplet for $g_0=-69$ and $g_1=155$ ($t\gtrsim 8000$ in Fig. 1) when the peak density becomes (a) maximal and (b) minimal. The dashed curves are the Gaussian functions fitted to the density profiles.