Stabilization of a matter-wave droplet in free space by feedback control of interatomic interactions

A Bose-Einstein condensate in three-dimensional free space is known to be unstable against expansion or collapse if the scattering length of atoms is constant. We propose a method to produce a stable self-trapped condensate in which nondestructive measurement of the condensate’s peak density is used to perform feedback control of the scattering length. The stability is shown to be robust against experimental imperfections in the measurements.

Figure 1

Stationary unstable solution of Eq. (8) (solid curve) for $g=-{\left(2\pi \right)}^{3∕2}$. The Gaussian function $e^{-r^2}∕{\pi }^{3∕2}$ is superimposed as a dashed curve for comparison. See Sec. 4 for the choice of parameters. The inset shows the same functions on a logarithmic scale to show the difference in the large-$r$ behavior between the stationary solution and the Gaussian function.

Figure 2

Time evolution of (a) the deviation $\tilde{D}\equiv D-D_0$ of the peak column density [see Eqs. (9, 10)] from the target value and (b) the fraction of remaining atoms $N∕N_0$ (solid curve) and the strength of the interaction $\tilde{g}=g-g_0$ (dashed curve) for $A=10$, $B=100$, $C=50$, $\Delta t=0.01$, $D_0=1∕\pi$, and $g_0=-{\left(2\pi \right)}^{3∕2}$. The initial state is the Gaussian wave function $\psi =e^{-r^2∕2}∕{\pi }^{3∕4}$, and the value of $\tilde{g}$ starts from 0 (not visible in the present time scale). The inset in (b) shows the stabilized density profile ${\mid \psi \mid }^2$ at $t=100$.

Figure 3

Dependence of the dynamics of $D$ on the feedback parameters $A$, $B$, and $C$. The other conditions are the same as those in Fig. 2.

Figure 4

Stability diagram of the BEC droplet under the feedback control for $A=10$. The other conditions are the same as those in Fig. 2. The open circles show stable points and the solid circles show unstable points. The dashed curve shows Eq. (23d).

Figure 5

Time evolution of the peak column density $D$ in the presence of measurement errors given in Eq. (24) with $\epsilon =0$ (solid line), $\epsilon =0.2$ (dashed line), $\epsilon =0.6$ (dotted line), and $\epsilon =0.7$ (dot-dashed line). The feedback parameters are $A=10$, $B=100$, and $C=0$. The other conditions are the same as those in Fig. 2.

Figure 6

Time evolution of the peak column density $D$ in the presence of finite spatial resolution given in Eq. (25) with $\sigma =0$ (solid line), $\sigma =0.2$ (dashed line), $\sigma =0.5$ (dotted line), and $\sigma =0.6$ (dot-dashed line). The feedback parameters are $A=10$, $B=100$, and $C=0$. The other conditions are the same as those in Fig. 2.