Droplet formation in a Bose-Einstein condensate with strong dipole-dipole interaction

Motivated by the recent experiment [H. Kadau et al., arXiv:1508.05007], we study roton instability and droplet formation in a Bose-Einstein condensate of ${}^{164}\mathrm{Dy}$ atoms with strong magnetic dipole-dipole interaction. We numerically solve the cubic-quintic Gross-Pitaevskii equation with dipole-dipole interaction, and show that the three-body interaction plays a significant role in the formation of droplet patterns. We numerically demonstrate the formation of droplet patterns and crystalline structures, decay of droplets, and hysteresis behavior, which are in good agreement with the experiment. Our numerical simulations provide the first prediction on the values of the three-body interaction in a ${}^{164}\mathrm{Dy}$ Bose-Einstein condensate. We also predict that the droplets remain stable during the time-of-flight expansion. From our results, further experiments investigating the three-body interaction in dipolar quantum gases are required.

Figure 1

Droplet patterns with droplet numbers $N_d$ ranging from 2 to 10. (a) Column density profiles at $t=10$–20 ms after a sudden quench of two-body interaction. In the insets, we take into account one micrometer resolution as in the experiment by taking the convolution of the column density profiles with the Gaussian with one micrometer width. The field of view is $11.5\times 11.5\mu \text{m}$ as well as in the insets. The unit of the density is $N/l_x^2$. (b) The typical isodensity surface with $N_d=8$. The color represents the phase at the surface. The size of the box is $11.5\times 11.5\times 5.8\mu \text{m}$.

Figure 2

Column density profiles in the dynamics of four-droplet formation. The number of atoms is $N=7500$. The unit of the density is $N/l_x^2$ and the field of view is $11.5\times 11.5\mu \text{m}$.

Figure 3

Relation between number of droplets $N_d$ and number of atoms $N$. We perform 280 calculations with different numbers of atoms and different initial noises for statistical analysis. The error bars show the standard deviation. The dashed line indicates $N_d=N/1755$.

Figure 4

Time evolution of the relative spectral weight (red triangles) and the number of atoms (gray solid line). The insets show the column density profiles at $t=0$, 20, 500, and 800 ms. The field of view in the insets is $11.5\times 11.5\mu \text{m}$.

Figure 5

Hysteresis of pattern formation. We prepare a stable state at $a=90a_0$, then perform time evolution with reducing $a$ to $85a_0$ in 4 ms. Subsequently, we keep $a=85a_0$ for 20 ms, then increase $a$ back to $90a_0$ in 4 ms and keep it for 20 ms. The insets show the column density profiles at $t=0,24$, and 40 ms, where the unit of the density is $N/l_x^2$ and the field of view is $11.5\times 11.5\mu \text{m}$.

Figure 6

Column density profiles in a time-of-flight simulation, where the trap potential is switched off at $t=10$ ms. The number of atoms is $N=15000$. The unit of the density is $N/l_x^2$ and the field of view is $25\times 25\mu \text{m}$.