Geometric filtering effect in expanding Bose-Einstein condensate shells

TL;DR

This work analyzes how the curved geometry of shell-shaped Bose-Einstein condensates shapes their free expansion, particularly regarding the fate of finite-angular-momentum components. By combining phase-imprinting and finite-temperature Bogoliubov approaches on a spherical shell, and validating with 3D Gross-Pitaevskii dynamics, the authors show that a radial centrifugal barrier on the sphere filters higher-$l$ modes, funneling population into the central peak for $l=0$ while higher-$l$ components are repelled or converted during inward focusing. The central density thus encodes the imprinting strength and the temperature, enabling a practical 2D thermometry method for spherical shells using simple absorption-imaging. The results illuminate a curvature-induced dynamic filtering mechanism in interacting quantum gases and are directly testable with current experimental platforms, with implications for shell thickness tuning and the study of nonlinear radial dynamics.

Abstract

A shell-shaped Bose-Einstein condensate released from its confinement expands radially both outwards and inwards, displaying a self-interference pattern characterized by a density peak surrounded by a halo. Here we analyze how an external imprinting or the thermal fluctuations of the condensate phase influence this expansion. In both cases, we find that the curved geometry filters the imploding finite angular-momentum modes via a radial centrifugal potential, so that only the condensate state can reach the origin and form the central peak. As a consequence, we observe a pronounced dependence of the central density on the imprinting strength and on temperature. This geometric filtering effect characterizes the free expansion of curved atomic gases in contrast with flat counterparts, it is easily observable in the available experimental platforms, and enables two-dimensional shells thermometry via simple absorption-imaging techniques.

Figures (4)

• Figure 1: (a): Maximal population ratio penetrating the radial region $r< R_{c}$ vs the phase-imprinting strength $C$. When increasing $C$ in the imprinted state $\Psi(\mathbf{r},t=0)$, the initial occupation $N_l/N$ of even-$l$ angular momentum states shifts toward larger $l$ values [see (b)], and large-$l$ modes are more strongly repelled from the region $r=0$ by the centrifugal barrier [see (c)]. Here we choose the following realistic parameters jia2022: $N=10^4$, $\gamma/E_R=7.7$, $l_0/R = 0.15$, and $R_{c}/R = 0.3$. The densities of panel (c) are obtained for $C=6$ and are reported at the time at which the central population is maximal, $t=0.15 t_R$.
• Figure 2: Time evolution of $N_0/N$, demonstrating the role of nonlinear interactions during the inward implosion of the condensate shell. Note how, due the coupling at the second line of Eq. \ref{['componentsGPE-1']}, the population tends to convert into the condensate as the shell expands and the central density increases. We use for this figure the same parameter and scales of Fig. \ref{['fig1']}.
• Figure 3: Hierarchy of energy scales identifying different regimes of trapped bosonic gases. We implement a 2D theory that includes both phase and density fluctuations to describe the free expansion of a two-dimensional shell-shaped Bose-Einstein condensate in the regime $\gamma \lesssim k_B T \ll \hbar\omega_0$.
• Figure 4: Maximal population ratio penetrating the radial region $r< R_{c}$ vs temperature. Similarly to the phase-imprinting case, larger temperatures produce condensate states with larger initial angular momenta, which are impeded to reach the central radial regions by the centrifugal potential repulsion. At a given temperature $T$, the maximal central population tends to decrease either when the shell thickens at fixed $N$, or when $N$ decreases at fixed shell thickness. Note that we consider here a thin shell in the 2D regime of Fig. \ref{['fig3']}, that the critical temperatures for $N = \{ 1 , 2 \} \times 10^4$ are $\{ 800, 1600\} \, k_B/E_R$, and we use for this figure $R_{c}/R = 0.3$. The error bars show the standard deviation obtained from averaging over many realizations of the initial state.