Numerical calculation of dipolar-quantum-droplet stationary states

We describe and benchmark a method to accurately calculate the quantum droplet states that can be produced from a dipolar Bose-Einstein condensate. Our approach also allows us to consider vortex states, where the atoms circulate around the long-axis of the filament-shaped droplet. We apply our approach to determine a phase diagram showing where self-bound droplets are stable against evaporation, and to quantify the energetics related to the fission of a vortex droplet into two nonvortex droplets.

Figure 1

Density profiles of (a) $s=0$ and (b) $s=1$ self-bound stationary states for a system with ${\epsilon }_{dd}^{-1}=0.5$ and $N={10}^4$ atoms. (c)–(e) Gradient flow evolution for case (b) of Table 2 using $a_{\mathrm{\Delta }t}=0.5$, showing (c) the error $r_{\infty }$, (d) time step $\mathrm{\Delta }{t}_n$, (e) energy error, and (f) the absolute virial expectation error. In the energy error $E_{\text{ref}}$ is a reference energy calculated from a state using a larger more dense grid.

Figure 2

(a) Phase diagram of self-bound $s=0$ and $s=1$ droplet solutions. Shaded regions bordered by black lines mark where $E_{\text{G}}<0$. Lines with circle markers show where the EGPE solution has $E=0$. (b) The energy and (c) $N\mu$ versus ${\epsilon }_{dd}^{-1}$ for $N={10}^4$ for the variational (lines) and GPE (lines with small circles) results for $s=0$ (red) and $s=1$ (blue). The dotted lines indicate where $E=0$. The inset shows the data near $E=0$. Subplots (d) and (e) show examples of metastable states with positive energies. (d) $s=0$ droplet with ${\epsilon }_{dd}^{-1}=0.75, E=1.7634\times {10}^{-2}, \mu =-2.1742\times {10}^{-5}$, and (e) $s=1, {\epsilon }_{dd}^{-1}=0.56, E=1.1746\times {10}^{-1}, \mu =-2.6444\times {10}^{-4}$.

Figure 3

Vortex droplet fission. (a) A schematic of the fission process where a $s=1$ vortex droplet (blue) decays into two $s=0$ droplets (red). (b) Fission energy $\mathrm{\Delta }{E}_N$ as a function of ${\epsilon }_{dd}^{-1}$ for $N={10}^4$ (black lines/markers) and $N={10}^5$ (red lines/markers). The results are calculated from the variational (lines) and EGPE (markers) theories. The vertical dotted line indicates where the $N={10}^{4}s=1$ droplet has zero-energy.