Merging and oscillations of dipolar Bose-Einstein condensate droplets

Abstract

We investigate the dynamics of Bose-Einstein condensate droplets composed of $^{164}$Dy atoms formed in a double-well potential following removal of the interwell barrier. By solving the dipolar Gross-Pitaevskii equation, we determine phase diagrams of ground-state configurations as functions of the atom number confined in the double-well potential. For strong dipolar interactions, some of the lowest-energy configurations arise from spontaneous symmetry breaking of the droplet structure, which optimizes the interaction energy. We analyze the subsequent time evolution after removal of the central barrier, revealing both droplet oscillations and merger events leading to the formation of larger droplets. The oscillations are driven by the external potential and by the repulsive tails of the in-plane component of the dipolar interaction. Merger events occur when the initial excess energy is sufficient to overcome the interdroplet potential barrier. The oscillatory dynamics depend sensitively on the atom number, the strength of dipolar interactions, and the initial symmetry of the configuration. We find that both oscillations of individual droplets and atom leakage from the droplets, induced by close droplet-droplet encounters, contribute to the damping of the oscillations.

Figures (19)

• Figure 1: The applied model potentials for the $y$ and $z$ directions with $\omega_y=60$Hz (red line) with $\omega_z=120$Hz (blue line). The potential for $x$ is plotted for a single (dashed black line) and double well (solid black line) for $2d=3\mu$m.
• Figure 2: (a) The contributions to the total energy for the ground-state at $\varepsilon_{dd}=1.5$ as a function of $N$ for the double well potential with minima spaced by $2d=3000$ nm; (b) maximal atom density at $z=0$ plane (red lines) and droplet height $H_z$ in $z$ direction (blue lines). The vertical lines mark the ranges of a varied localization type: I -- a delocalized system (no droplet formation), II -- an asymmetric system with fully developed droplet in one of the wells, III -- a symmetric system with droplet in both wells, IV -- two droplets in one well and one in the other, V -- a symmetrical system with 4 droplets. (c-h) Isosurfaces of the BEC density for $N=10^3$ (c), $6\times 10^3$ (d), $8\times 10^3$ (e), $2.7\times 10^4$ (f), $3.8\times 10^4$ (g) and $4\times 10^4$ (h) in the lowest-energy configuration. The isosurfaces correspond to 20% (transparent) and 80% (opaque) of the maximum density value. The plots span the area of $x$ (horizontal direction) and $y$ from -3.6 $\mu$m to 3.6 $\mu$m. At the basis of the plots, below the isosurfaces, a cross-section of the BEC density at $z=0$ is presented. The colorscale for the density is given in $\mu$m$^{-3}$ units. The plots correspond to regions marked by I (c), II (d), III (e,f), IV (g) and V (h).
• Figure 3: (a) Total energy and contributions to the energy for the ground-state at $\varepsilon_{dd}=1.5$ as a function of $N$ for external potential with a single minimum. (b) the maximal gas density at $z=0$ plane (red lines) and droplet height $H_z$ in $z$ direction (blue lines). The vertical lines mark the ranges of a varied localisation type. I -- a delocalized system, II -- a single droplet in the center of the system, III -- two droplets, IV -- three droplets, V -- four droplets. (c-h) Isosurfaces of the BEC density for $N=5\times 10^3$ (c), $6\times 10^3$ (d), $7.5\times 10^3$ (e),$2\times 10^4$ (f), $2.75\times 10^4$(g) and $4\times 10^4$ (h) for $\varepsilon=1.5$ and the lowest-energy configuration for a potential with a single minimum. The isosurfaces correspond to 20% (transparent) and 80% (opaque) of the maximum density. The plots span the area of $x$ (horizontal direction) and $y$ from -3.6 $\mu$m to 3.6 $\mu$m. At the basis of the plot, below isosurfaces a cross-section of the BEC density at $z=0$ is presented. The colorscale for the density is given in $\mu$m$^{-3}$ units. The plots (c-h) correspond to regions marked by I, II, II (still), III, IV and V in (a) and (b).
• Figure 4: Snapshots of the time evolution of the BEC density for $d=1500$ nm $\varepsilon_{dd}=1.5$ for $N=2500$ atoms. The plots show the cross section taken at $z=0$ with $x\in[-4\mu$m,$4\mu$m] and $y\in[-4\mu$m,$4\mu$m]. The colorscale for the density is given in $\mu$m$^{-3}$ units. Plots (a-j) show the moments $t$ after the confinement potential is set to a single well, with $t=0$ (a), 2.42 ms (b), 4.84 ms (c), 7.26 ms(d), 9.68 ms(e), 12.1 ms (f), 14.51 ms(g), 16.93 ms(h), 19.35 ms(i). (j) Contributions to the energy per atom as functions of time.
• Figure 5: Snapshots of the time evolution of the BEC density prepared for $d=1500$ nm $\varepsilon_{dd}=1.5$ for $N=5000$ atoms. The plots show the cross section taken at $z=0$ with $x\in[-4\mu$m,$4\mu$m] and $y\in[-4\mu$m,$4\mu$m]. The color scale for the density is given in units of $1000\mu$m$^{-3}$. Plots (a-l) show the moments $t$ after the confinement potential is set to a single well, with $t=0$ (a), 2.42 ms (b), 4.84 ms (c), 7.26 ms(d), 9.68 ms(e), 12.1 ms (f), 14.51 ms(g), 16.93 ms(h), 19.35 ms(i) 21.77 ms(j), 24.19 (k), 26.61 (l). (m) Contributions to the energy per atom as functions of time.