Shell-shaped Bose-Einstein condensates: Dynamics, excitations, and thermodynamics

Abstract

Shell-shaped Bose-Einstein condensates (BECs) represent a paradigmatic instance of quantum fluids in hollow geometries exhibiting phenomena that bridge from ultracold atomic to astrophysical realms. In this work, we present a comprehensive survey of the dynamics, thermodynamics, and collective excitations of shell-shaped BECs, synthesizing two decades of our group's theoretical work in light of recent experimental breakthroughs. We begin by analyzing the evolution of a BEC from filled-sphere to hollow-shell geometries, illustrating the necessity of microgravity conditions to avoid gravitational sag. We then analyze collective modes structure across this evolution and pinpoint a universal dip in the frequency spectra as well as mode reconfiguration due to inner-surface excitations as robust signatures of the hollowing-out transition. Turning to vortex physics, we show that the closed-surface topology enforces vortex-antivortex configurations in shell-shaped BECs and that the natural annihilation of these pairs can be stabilized by rotation, with the critical rotation rate depending on shell thickness. In the thermodynamic domain, we investigate the interplay between shell inflation and the BEC phase transition, where adiabatic expansions lead to condensate depletion. This behavior motivates a study of the nonequilibrium dynamics of shell-shaped BECs; we perform such a study by constructing a time-dependent dynamic technique that can capture the evolution in both adiabatice and non-adiabatic regimes. Finally, we review recent experimental realizations of shell-shaped BECs, including the landmark creation of ultracold shells aboard the International Space Station, and outline prospects for exploring quantum fluids in curved geometries.

Figures (10)

• Figure 1: (Color online) Schematic density profiles $n_{\textrm{eq}}(\mathbf{r})$ of a Bose-Einstein condensate evolving from filled-sphere to hollow-shell geometries. (Adapted from Ref. Sun2018. Copyright (2018) by the American Physical Society.)
• Figure 2: (Color online) (a--d) Equilibrium density profiles $n_{\textrm{eq}}(r)$ (thin curves; with the maximum value set to 1) of a Bose-Einstein condensate and the corresponding bubble-trap potential (thick curves; in arbitrary units) for various detuning $\Delta$, showing the evolution from the filled-sphere geometry at $\Delta=0$ to a hollow thin-shell one at a large $\Delta$. The solid profiles show the numerical solutions of the GP equation, while the dashed ones are based on the Thomas-Fermi approximation.
• Figure 3: (Color online) Thomas-Fermi density profiles in the $x$-$z$ plane for condensates confined by a bubble trap without gravity (left) and under the influence of gravitational fields $0.0017g$ (middle) and $0.007g$ (right), where $g$ is the gravitational acceleration on Earth (in the $-z$ direction). These profiles are generated for $10^5$$^{87}$Rb atoms in a bubble trap approximated by Eq. (\ref{['eq:off_center_harm']}) with $\omega_{\rm{sh}}=403$ Hz, forming a condensate shell with outer radius $20$$\mu$m and thickness $4$$\mu$m in the absence of gravity. The colors in the bar graph represent density normalized by $n_m=2.96 \times 10^{13}/\textrm{cm}^{3}$. 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.
• Figure 4: (Color online) Oscillation frequencies $\omega$ of the three lowest-lying ($\nu=1,2,3$) spherically symmetric ($l=0$) collective modes vs the normalized bubble-trap detuning $\tilde{\Delta}$ ($\equiv \Delta/R^2$). The zero frequency curve ($\nu=0$ mode) is presented for comparison. The condensate evolves from a filled sphere ($\tilde{\Delta}=0$) toward a hollow thin shell ($\tilde{\Delta} \to 1$), through a hollowing transition at $\tilde{\Delta}=0.5$, where the frequency curves exhibit a dip due to the appearance of a sharp new boundary in the density profile. Insets: schematic density oscillation profiles $\delta n(r)$ of the corresponding collective modes of the filled sphere and hollow shell condensates. (Adapted from Ref. Sun2018. Copyright (2018) by the American Physical Society.)
• Figure 5: (Color online) Oscillation frequencies $\omega$ of the four lowest-lying ($\nu=0,1,2,3$) large-angular-momentum ($l=20$) collective modes vs the bubble-trap detuning $\tilde{\Delta}$, with the same convention as in Fig. \ref{['fig:collective_fig_1']}. Insets: schematic density oscillation profiles $\delta n(\mathbf{r})$ showing one outer-surface mode on the filled condensate and two inner-surface and outer-surface modes on the shell-shaped condensate. (Adapted from Ref. Sun2018. Copyright (2018) by the American Physical Society.)