Quantum Bubbles in Microgravity

The recent developments of microgravity experiments with ultracold atoms have produced a relevant boost in the study of shell-shaped ellipsoidal Bose-Einstein condensates. For realistic bubble-trap parameters, here we calculate the critical temperature of Bose-Einstein condensation, which, if compared to the one of the bare harmonic trap with the same frequencies, shows a strong reduction. We simulate the zero-temperature density distribution with the Gross-Pitaevskii equation, and we study the free expansion of the hollow condensate. While part of the atoms expands in the outward direction, the condensate self-interferes inside the bubble trap, filling the hole in experimentally observable times. For a mesoscopic number of particles in a strongly interacting regime, for which more refined approaches are needed, we employ quantum Monte Carlo simulations, proving that the nontrivial topology of a thin shell allows superfluidity. Our work constitutes a reliable benchmark for the forthcoming scientific investigations with bubble traps.

Figure 1

Critical temperature for Bose-Einstein condensation $T_{\mathrm{BEC}}$ as a function of the number of particles $N$ (top) and detuning $\mathrm{\Delta }$ (bottom). Top: comparison for different external potentials: harmonic trap with $\overrightarrow{\omega }/(2\pi )=(30,100,100)\mathrm{Hz}$ (red thin line), noninteracting bosons in a harmonic trap [40] (blue dot-dashed line), bubble trap with $\mathrm{\Delta }/(2\pi )=10\mathrm{kHz}$ (grey dashed line), bubble trap with $\mathrm{\Delta }/(2\pi )=30\mathrm{kHz}$ (green thick line). Bottom: as soon as $\mathrm{\Delta }$ is nonzero $T_{\mathrm{BEC}}$ decreases partly due to the reduced maximal local density, becoming essentially constant for large detunings [37].

Figure 2

Top: contour plot of the density in the $xz$ plane (colorbox units in $\mu {\mathrm{m}}^{-3}$), obtained solving Eq. (2), for $\overrightarrow{\omega }/(2\pi )=(30,100,100)\mathrm{Hz}$, $\mathrm{\Delta }/(2\pi )=30\mathrm{kHz}$, $\mathrm{\Omega }/(2\pi )=5\mathrm{kHz}$, and $N=57100$. Bottom: one-dimensional sections of the density at $T=0$ (from the GPE) and at $T_{\mathrm{BEC}}$ (from Hartree-Fock theory). Note that at $T=0$ the condensate is concentrated on the shell vertices, while the thermal cloud at $T_{\mathrm{BEC}}$ is uniformly distributed.

Figure 3

Free expansion of the condensate shell, initially in the ground state for the bubble-trap potential of Eq. (1) with $\mathrm{\Delta }/(2\pi )=10\mathrm{kHz}$, and the other parameters as in Fig. 2. From left to right and from top to bottom, the condensate slices along the $xz$ plane are taken at the times: 0 ms, 4.5 ms, 9 ms, and 18 ms. The hollow condensate expands both outwards and inwards, showing a qualitatively different interference pattern with respect to that of harmonic traps [37, 46].

Figure 4

Results of the Monte Carlo simulations, in which we employ the bubble-trap potential of Eq. (1) with $\overrightarrow{\omega }/(2\pi )=(0.2,1,1)\mathrm{kHz}$, $\mathrm{\Delta }/(2\pi )=10\mathrm{kHz}$, $\mathrm{\Omega }/(2\pi )=5\mathrm{kHz}$, and $a_s=50a_{\mathrm{Rb}}$. In (a) we represent the real space projection of the wordlines for $N=512$ bosons at 4.4 nK. The superfluid character of this configuration is proven in panel (b), by the fat-tailed distribution $P(n)$ of the $n$-particles permutation cycles, with $1\le n\le N$. We summarize our simulations in panel (c), showing the superfluid fraction of the system $f_s^{(x)}$ as a function of the temperature $T$.