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Geometric potential for a Bose-Einstein condensate on a curved surface

Sheilla M. de Oliveira, Natália Salomé Móller

TL;DR

This work derives a curvature-induced geometric potential for a Bose-Einstein condensate confined to a curved surface by performing a perturbative expansion and dimensional reduction from a 3D Gross-Pitaevskii framework to a 2D equation on the surface. For a prolate ellipsoid, the geometric potential drives higher density at the equator, linking ground-state structure to surface geometry through the term proportional to $( kappa_1- kappa_2)^2$. The analysis clarifies discrepancies with bubble-trap experiments by showing that pole localization can arise from confinement asymmetries rather than intrinsic curvature, and demonstrates that equator localization is a geometry-driven effect accessible to experiments with properly engineered confinement. The results offer a tractable, geometry-centered lens for predicting and interpreting BEC behavior on curved manifolds and can be extended to broader geometries and nontrivial surfaces.

Abstract

We compute the ground state of a Bose-Einstein condensate confined on a curved surface and unravel the effects of curvatures. Starting with a general formulation for any smooth surface, we apply it to a prolate ellipsoid, which is inspired by recent bubble trap experiments. Using only elementary tools, we perform a perturbative approach to the Gross-Pitaevskii equation and a general Ansatz, followed by a dimensional reduction. We derive an effective two-dimensional equation that includes a curvature-dependent geometric potential. We compute the ground state using Thomas-Fermi approximation and, for an isotropic confinement, we find that the highest accumulation of atoms happens on the regions with the greatest difference between the principal curvatures. For a prolate ellipsoid, this accumulation happens on the equator, which is contrary to previous findings that describe accumulation on the poles of a bubble trap. Finally, we explain the reasons for this difference: the higher accumulation of atoms on the poles happens due to anisotropies in the confinement, while the higher accumulation on the equator happens exclusively due to the geometric properties of the surface.

Geometric potential for a Bose-Einstein condensate on a curved surface

TL;DR

This work derives a curvature-induced geometric potential for a Bose-Einstein condensate confined to a curved surface by performing a perturbative expansion and dimensional reduction from a 3D Gross-Pitaevskii framework to a 2D equation on the surface. For a prolate ellipsoid, the geometric potential drives higher density at the equator, linking ground-state structure to surface geometry through the term proportional to . The analysis clarifies discrepancies with bubble-trap experiments by showing that pole localization can arise from confinement asymmetries rather than intrinsic curvature, and demonstrates that equator localization is a geometry-driven effect accessible to experiments with properly engineered confinement. The results offer a tractable, geometry-centered lens for predicting and interpreting BEC behavior on curved manifolds and can be extended to broader geometries and nontrivial surfaces.

Abstract

We compute the ground state of a Bose-Einstein condensate confined on a curved surface and unravel the effects of curvatures. Starting with a general formulation for any smooth surface, we apply it to a prolate ellipsoid, which is inspired by recent bubble trap experiments. Using only elementary tools, we perform a perturbative approach to the Gross-Pitaevskii equation and a general Ansatz, followed by a dimensional reduction. We derive an effective two-dimensional equation that includes a curvature-dependent geometric potential. We compute the ground state using Thomas-Fermi approximation and, for an isotropic confinement, we find that the highest accumulation of atoms happens on the regions with the greatest difference between the principal curvatures. For a prolate ellipsoid, this accumulation happens on the equator, which is contrary to previous findings that describe accumulation on the poles of a bubble trap. Finally, we explain the reasons for this difference: the higher accumulation of atoms on the poles happens due to anisotropies in the confinement, while the higher accumulation on the equator happens exclusively due to the geometric properties of the surface.
Paper Structure (17 sections, 105 equations, 5 figures)

This paper contains 17 sections, 105 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries
  3. Model
  4. Perturbative expansion
  5. Dimensional reduction
  6. Thomas Fermi approximation
  7. Ellipsoid
  8. Ground state and comparison to the bubble trap
  9. Discussion
  10. Conclusions
  11. Acknowledgements
  12. Justification for the term $\mu_1=0$
  13. Gaussian distributions and Dirac delta function
  14. Justification for the vanishing of $S_2$-involved terms after integration
  15. The Gaussian normal coordinate system for a prolate ellipsoid
  16. ...and 2 more sections

Figures (5)

  • Figure 1: Geometric potential on ellipsoids for different values of $a$ and $b$.
  • Figure 2: Profile of the gas density on the ellipsoid surface for different values of $\beta$.
  • Figure 3: Strip on an ellipsoid, being definined by $\varphi\in [(\pi-\Delta\varphi)/2,(\pi+\Delta\varphi)/2]$ and $\theta\in[0,2\pi)$.
  • Figure 4: a) Number of atoms for different values of $\beta$ in the interval $\Delta\varphi$. b) Difference on the atom number between $\beta=0$ and $\beta'$ in the interval $\Delta\varphi$, for different values of $\beta'$.
  • Figure 5: Cartesian, prolate spheroidal and Gaussian normal coordinate systems represented on the plane with constant $\theta$ for the points ${\bf q}$ and ${\bf p}$. The point ${\bf p}$ lies on a prolate ellipsoid and is the nearest point on this surface to the point ${\bf q}$.