Observation of Quantum Droplets in a Strongly Dipolar Bose Gas

Quantum fluctuations are the origin of genuine quantum many-body effects, and can be neglected in classical mean-field phenomena. Here, we report on the observation of stable quantum droplets containing $\sim 800$ atoms that are expected to collapse at the mean-field level due to the essentially attractive interaction. By systematic measurements on individual droplets we demonstrate quantitatively that quantum fluctuations mechanically stabilize them against the mean-field collapse. We observe in addition the interference of several droplets indicating that this stable many-body state is phase coherent.

Figure 1

Quantum droplets of a dipolar Bose gas in a waveguide. (a) Schematic representation of the droplets in the waveguide; the elongation along $z$ is represented; their separation $d$ is indicated. (b) Examples of in situ optical density (OD) images after release in the waveguide at the magnetic field $B_1=6.656(10)\mathrm{G}$. Images taken at times $t_{\mathrm{WG}}=0$, 5, 10, 15, 20 ms (top to bottom). The OD is normalized to the maximal OD in each image to improve visibility. (c) Evolution of the mean separation $d$ between the droplets as a function of time. (d) Blue circles: evolution of the width $\sigma$ obtained from a Gaussian fit to their density profiles (average of transverse and axial radii). Red diamonds: evolution of the size of a BEC for comparison. The data in panels (c) and (d) are obtained by averaging at least four experimental realizations; the error bars indicate the statistical standard deviation The convention for the axes used through the Letter is indicated in panel (a).

Figure 2

Derivative of the chemical potential with respect to density as a function of density, at the center of a droplet using a Gaussian ansatz (1) ($g=4\pi {\hslash }^{2}a/m$). The blue shaded region expresses our uncertainty on the scattering length. Negative values imply mechanical instability. The experimental value obtained from expansion measurements (Fig. 4) is shown as a red circle assuming a Gaussian distribution and as an orange square assuming an inverted parabola. The dashed line shows the same quantity obtained using a three-body repulsion using parameters from Ref. [22], which stabilizes at a higher density.

Figure 3

Ratio of the lifetime ${\tau }_f/\tau i$ of the droplets between scattering lengths $a_f$ and $a_i$. We use here $a_i=94(12)a_0$ obtained at $B_i=6.573(5)\mathrm{G}$. The data points are taken down to $B_f=6.159(5)\mathrm{G}$. The filled blue and green hatched areas represent the expected scaling using quantum fluctuations and three-body repulsion, respectively, taking into account the uncertainty range on the droplets’ aspect ratio: $0\le \kappa \le 0.2$.

Figure 4

Time-of-flight expansion measurements. The field is held at $B_1=6.656(10)\mathrm{G}$ until release when it is quenched to a different value. (a),(b) Images where the field is kept at $B_1$ during expansion and (c),(d) quenched to 6.86 G. In (a) and (b) one sees expanding droplets, whereas in (c) and (d) they overlap and clear interference fringes appear along the $x$ axis while we can still measure the expansion size in the $y$ direction.