Static and dynamic properties of shell-shaped condensates

Static, dynamic, and topological properties of hollow systems differ from those that are fully filled as a result of the presence of a boundary associated with an inner surface. Hollow Bose-Einstein condensates (BECs) naturally occur in various ultracold atomic systems and possibly within neutron stars but have hitherto not been experimentally realized in isolation on Earth because of gravitational sag. Motivated by the expected first realization of fully closed BEC shells in the microgravity conditions of the Cold Atomic Laboratory aboard the International Space Station, we present a comprehensive study of spherically symmetric hollow BECs as well as the hollowing transition from a filled sphere BEC into a thin shell through central density depletion. We employ complementary analytic and numerical techniques in order to study equilibrium density profiles and the collective mode structures of condensate shells hosted by a range of trapping potentials. We identify concrete and robust signatures of the evolution from filled to hollow structures and the effects of the emergence of an inner boundary, inclusive of a dip in breathing-mode-type collective mode frequencies and a restructuring of surface mode structure across the transition. By extending our analysis to a two-dimensional transition of a disk to a ring, we show that the collective mode signatures are an essential feature of hollowing, independent of the specific geometry. Finally, we relate our work to past and ongoing experimental efforts and consider the influence of gravity on thin condensate shells. We identify the conditions under which gravitational sag is highly destructive and study the mode-mixing effects of microgravity on the collective modes of these shells.

Figure 1

Top row: schematic equilibrium density profiles of a spherical BEC evolving from filled (left two panels) to hollow (right two) geometries. The middle two columns are close to the hollowing transition. Second and third rows: density deviation of collective modes $(\nu ,\ell )=(1,0)$ and (1,1), respectively, for the corresponding equilibrium density profile in the same column. Bottom row: density deviation of high angular momentum ($\ell =20$) surface modes on a filled BEC (left, on the outer surface only) and a hollow BEC (right, on either the inner or outer surface). The density and density deviations are scaled by colors in the corresponding bar graph.

Figure 2

Ground-state density profiles $n_{\mathrm{eq}}\left(r\right)$ (multiplied by 1000) in the bubble trap for $\mathrm{\Delta }=0$ (lighter blue curves), 25 (light green), 50 (dark red), 100 (darker gray), and 200 (black). The various values of $\mathrm{\Delta }$ show the evolution from the spherical (filled) condensate in a harmonic trap at $\mathrm{\Delta }=0$ to a thin-shell condensate for large $\mathrm{\Delta }$ (the 3D visualization can be found in the top row of Fig. 1). The solid (dashed) curves are obtained from the numerical solution of the GP equation (the Thomas-Fermi approximation). Here, we use $\mathrm{\Delta }/\mathrm{\Omega }=1$ and an interaction constant $u=10000$.

Figure 3

Oscillation frequencies $\omega$, found via the hydrodynamic equation, Eq. (10), with Thomas-Fermi equilibrium profiles, of the three lowest lying nonzero spherically symmetric ($l=0$) collective modes $\nu =1,2,3$ (dark red, green, and light blue curves, respectively) vs the bubble-trap detuning $\tilde{\mathrm{\Delta }}$ [given $\mathrm{\Delta }/\mathrm{\Omega }=1$ in Eq. (4)]. The zero mode (black) is also presented for comparison. As $\tilde{\mathrm{\Delta }}$ increases, the BEC evolves from a filled sphere $\tilde{\mathrm{\Delta }}=0$ toward a hollow thin shell $\tilde{\mathrm{\Delta }}\to 1$, through a hollowing transition at $\tilde{\mathrm{\Delta }}=0.5$. In the sphere and thin-shell limits, the frequencies agree with the exact solutions of Eqs. (16) and (20), respectively. Around the transition point, the collective-mode evolution is characterized by a dip in frequency, which is singular in the Thomas-Fermi approximation due to the appearance of a sharp new boundary.

Figure 4

Normalized density deviation profiles $\delta n\left(r\right)$ of the three physical breathing modes in Fig. 3 (same convention) and equilibrium profile $n_{\mathrm{eq}}\left(r\right)$ (dashed curves). Individual panels correspond to three filled cases $\tilde{\mathrm{\Delta }}=0$, 0.4, and 0.499, and three hollow cases 0.501, 0.6, and 0.99 (as labeled inside). Around the hollowing-out transition, the density deviation profiles tend to concentrate on the spherical center. This is energetically favorable and related to the development of frequency dip, as discussed in Sec. 6b.

Figure 5

Oscillation frequencies $\omega$, found via the hydrodynamic equation, Eq. (9), with equilibrium profiles given by the numerical solution to the GP equation, of the three lowest-lying nonzero spherically symmetric ($l=0$) collective modes $\nu =1,2,3$ (circles, squares, and triangles, respectively) vs the bubble-trap detuning $\mathrm{\Delta }$. The bubble-trap parameters are set by $\mathrm{\Delta }/\mathrm{\Omega }=1$ and the interaction strength is $u={10}^4$. The equilibrium profiles corresponding to five of the data points are shown in Fig. 2. Compared to the Thomas-Fermi results in Fig. 3, the dip in frequency is softened but clearly presents a hollowing transition region (shaded region) between filled and hollow topologies.

Figure 6

Oscillation frequencies $\omega$, found via quench numerics, of the three lowest-lying nonzero spherically symmetric ($l=0$) collective modes $\nu =1,2,3$ (same convention as Fig. 5) vs the bubble-trap detuning $\mathrm{\Delta }$. The bubble-trap parameters are set by $\mathrm{\Delta }/\mathrm{\Omega }=1$ and the interaction strength is $u={10}^4$. In going beyond the Thomas-Fermi approximation of Fig. 5, the singularity in the dip feature around the hollowing transition is softened and spread across the shaded region. However, the drop in frequency from the filled $\mathrm{\Delta }=0$ point and the asymptote to the thin-shell limit for large $\mathrm{\Delta }$ persists and the spectrum demonstrates that the collective-mode features through the topological transition are robust beyond the hydrodynamic approximation.

Figure 7

Density deviation profiles $\delta n\left(r\right)$ of the three breathing modes in Fig. 6 (same convention) and equilibrium profile $n_{\mathrm{eq}}\left(r\right)$ (dashed curves). Individual panels correspond to $\mathrm{\Delta }=0$, 25, 50, 60, 70, and 200 (as labeled inside). The radial coordinate ($x$ axis) is presented in the same units as Fig. 2. Similar to the hydrodynamic results in Fig. 4, the density deviation profiles tend to concentrate (but less significantly so) around the spherical center in the hollowing transition region.

Figure 8

(a) [(b)] Oscillation frequencies $\omega$ vs $\tilde{\mathrm{\Delta }}$ for $\ell =1$ ($\ell =20$) and $\nu =0,1,2,3$ (same convention as Fig. 3). The $\ell =1$ (low-$\ell$) modes exhibit similar features, including the dip across the hollowing transition $\tilde{\mathrm{\Delta }}=0.5$, as the $\ell =0$ modes in Fig. 3, except that the $\nu =0$ mode here has nonzero frequency. The $\ell =20$ (high-$\ell$) modes exhibit sudden drops at the transition. (c) Density deviation profiles $\delta n$ corresponding to one mode $(\tilde{\mathrm{\Delta }},\nu )=(0.49,0)$ (dark red) right before the hollowing (axis on bottom), and two modes (0.51,0) (green) and (0.51,1) (light blue) right after it (axis on top) in panel (b). (d) Same convention as panel (c) except for modes (0.49,1) right before the hollowing and (0.51,2) and (0.51,3) right after it. Panels (c) and (d) confirm that the high-$\ell$ modes are surface modes—with density deviation localizing around the condensate surfaces. Before the hollowing ($\tilde{\mathrm{\Delta }}=0.49$), the density deviations of all modes localize around the only surface $r=R$, while after the hollowing ($\tilde{\mathrm{\Delta }}=0.51$), half of them remain around the outer surface $r=R$ and the other half redistribute around the emerging inner surface $r=R_{\mathrm{in}}\approx 0$.

Figure 9

(a) Thomas-Fermi inner boundary $R_{\mathrm{in}}^{\mathrm{gt}}$ (dark red) and maximum-density position $r_0^{\mathrm{gt}}$ (green) vs $\gamma$ for a condensate in the general trap, given by Eq. (5) for $\alpha =2$. Insets from left to right: schematic density profiles of the condensate at $\gamma =0$, 0.3, 0.5, and 0.8, respectively. (b) Thomas-Fermi density profiles $n_{\mathrm{eq}}^{\mathrm{gt}}\left(r\right)$ around the center for $\alpha =1$ (dark red), 1.5 (green), and 2 (light blue) at the hollowing transition $\gamma =0.5$.

Figure 10

(a) Oscillation frequencies $\omega$ for the lowest nonzero breathing mode ($\nu =1$, $\ell =0$) as a function of $\gamma$ for $\alpha =2$ (dark red), 1.7 (green), and 1.4 (light blue), obtained by solving the hydrodynamic equation for the general-trap Thomas-Fermi profiles in Eq. (27). (b) The corresponding density deviation profiles $\delta n$ at the hollowing transition, $\gamma =0.5$. (c) Oscillation frequencies $\omega$ obtained from the variational method. (d) Variational frequency function at $\gamma =0.5$. Panels (b)–(d) have the same convention as panel (a). Panels (a) and (b) show the dependence of the frequency dip sharpness and the concentration of density deviation profiles on the growth rate of the equilibrium density profile from the hollowing-out center. Panels (c) and (d) show that the dependence originates from the orthonormality and energy minimization of collective mode.

Figure 11

(a) [(b)] Oscillation frequencies $\omega$ of high-angular-momentum modes $\ell =20$ in the general-trap case vs $\gamma$ for $\nu =0,1,2,3$ (same convention as Fig. 3) and $\alpha =2$ (1.85). Comparing (a) and (b), we see that the sudden drop of surface-mode frequency is a universal feature for the hollowing transition, but the degeneracy between outer and inner surface modes of hollow condensates only occurs at $\alpha =2$.

Figure 12

(a) [(b)] Oscillation frequencies $\omega$ of a 2D condensate vs 2D bubble-trap parameter $\tilde{\mathrm{\Delta }}$ for breathing modes $\ell =0$ (edge modes $\ell =20$) and $\nu =0,1,2,3$ (same convention as Fig. 3). Both panels show the same collective-mode features in the filled-to-hollow evolution and upon the hollowing transition as in 3D cases, indicating that the hollowing-out physics is dimension independent.

Figure 13

Thomas-Fermi density profiles for condensates confined by the bubble trap without gravity (left) and under the influence of gravitational fields $0.0014g$ (middle) and $0.007g$ (right), where $g$ is the gravitational acceleration on Earth. These profiles are generated for ${10}^5{}^{87}\mathrm{Rb}$ atoms forming a condensate shell with outer radius $20\mu \mathrm{m}$ and thickness $4\mu \mathrm{m}$ in the absence of gravity. The colors in the bar graph represent density normalized by $n_M=3.1\times {10}^{13}/{\mathrm{cm}}^3$. As the strength of the gravitational field increases, we observe a density depletion at the top of the condensate shell and a density maximum at its bottom.