Bose-Einstein condensate in a shell: nonlinear modes and symmetry breaking

Abstract

The state of a quantum particle confined to a spherical surface is commonly described using spherical harmonics, which play a crucial role in solving many scientific problems featuring spherical symmetry. A mean-field description of a system of many interacting quantum particles confined to a spherical shell naturally leads to nonlinear generalizations of spherical harmonics. In this study, we analyze the emergence of such nonlinear modes in a shell-shaped Bose-Einstein condensate with both repulsive and attractive two-particle interactions. We find several classes of nonlinear modes. The first kind is an analytical continuation of a single spherical harmonic or their superposition for nonvanishing atom-atom interaction. Some of these solutions have planar counterparts in the form of Townes solitons and Townes vortices, which either appear stable even in the case of attractive interactions or can be stabilized by three-particle interactions. The second kind originate from symmetry-breaking bifurcations in the case of a repulsive interaction, and therefore have no linear counterparts. Moreover, we demonstrate the existence of a breathing dynamics that oscillates primarily between an unstable soliton and a stable vortex necklace state. Our findings show that the physics of nonlinear matter waves on curved geometries differs substantially from that in planar systems, suggesting the feasibility of observing these unconventional wave patterns experimentally.

The paper studies how interactions modify spherical-harmonic modes of a Bose–Einstein condensate confined to a thin spherical shell, and which stationary patterns and dynamics arise through symmetry breaking. It uses the angular Gross–Pitaevskii equation $i\partial_t\psi=\tfrac12L^2\psi+G|\psi|^2\psi$, with unit normalization and $G\approx4\pi aN/\Delta R$. Stationary states are computed by nonlinear eigenvalue iteration, stability by Bogoliubov–de Gennes spectra, and dynamics by direct simulations and spherical-harmonic Galerkin truncations. Nonlinear continuations of individual harmonics and selected degenerate superpositions exhibit stability windows for repulsive and attractive interactions. The homogeneous state has an analytical excitation spectrum and is dynamically stable for $G>-2\pi$. At $G=-2\pi$, a localized branch breaks rotational symmetry and approaches the planar Townes-soliton norm $|G|\approx5.85$. Attractive localized solitons and vortices are unstable in the cubic model but can be stabilized by repulsive quintic terms; the reported vortex example requires $\delta=30$. For repulsion, symmetry-breaking branches form vortex necklaces: the branch from the $l=1,m=0$ dark soliton appears near $G=26.3$ and has an extended stability interval. At $G=40$, an unstable dark soliton repeatedly evolves into a four-vortex configuration and approximately returns. Three spherical harmonics contain approximately 95% of the population; few-mode models reproduce the dynamics qualitatively but develop timing errors. The treatment assumes frozen radial motion, requiring interaction energies below the radial excitation gap. Mode families are explored selectively, largely through $l\leq4$. Experimental preparation routes are proposed, but no experimental verification is reported.