Bose-Einstein Condensation on the Surface of a Sphere

Motivated by the recent achievement of space-based Bose-Einstein condensates (BEC) with ultracold alkali-metal atoms under microgravity and by the proposal of bubble traps which confine atoms on a thin shell, we investigate the BEC thermodynamics on the surface of a sphere. We determine analytically the critical temperature and the condensate fraction of a noninteracting Bose gas. Then we consider the inclusion of a zero-range interatomic potential, extending the noninteracting results at zero and finite temperature. Both in the noninteracting and interacting cases the crucial role of the finite radius of the sphere is emphasized, showing that in the limit of infinite radius one recovers the familiar two-dimensional results. We also investigate the Berezinski-Kosterlitz-Thouless transition driven by vortical configurations on the surface of the sphere, analyzing the interplay of condensation and superfluidity in this finite-size system.

Figure 1

Top: Critical temperature for Bose-Einstein condensation in terms of the product $n{R}^2$, where $R$ is the radius of the sphere and $k_B{T}_{\mathrm{BEC}}$ is rescaled with the energy $\zeta ={\hslash }^{2}n/m$. Notice how our result in the semiclassical approximation (solid line) converges for $n{R}^2\gg 1$ to the numerical evaluation of the summation of Eq. (3) (dashed line). As expected, for fixed $n$ the critical temperature tends to zero in the thermodynamic two-dimensional limit $n{R}^2\to \infty$. Bottom: Condensate fraction $n_0/n$ for noninteracting bosons on the sphere surface in terms of the rescaled temperature $T/T_{\mathrm{BEC}}$, obtained from the numerical solution of Eq. (6) with $n{R}^2$ fixed.

Figure 2

Phase diagram of the system for two values of $n{R}^2$: $10^2$ in the upper panel, $10^4$ in the lower panel. The dashed lines represent the critical temperature $k_B{T}_{\mathrm{BEC}}/\zeta$, rescaled with the typical energy $\zeta ={\hslash }^{2}n/m$, plotted in terms of the adimensionalized zero-range interaction strength $gm/{\hslash }^2$. The solid curve represents the critical temperature $k_B{T}_{\mathrm{BKT}}/\zeta$ of the Berezinski-Kosterlitz-Thouless transition. Note that, depending on the values of $n{R}^2$ and of $gm/{\hslash }^2$ and within the approximations adopted (see text), the system can show coexistence of condensation and superfluidity ($\mathrm{BEC}+\mathrm{SF}$), or condensation in the absence of superfluidity (BEC).