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arXiv:2504.20751 · AVS Quantum Science

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

Sheilla M. de Oliveira, Natália S. Móller

Published Sep 1, 2025• 1 citation
arXivHTMLPDF

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 the Thomas–Fermi approximation and, for an isotropic confinement, we find that the highest accumulation of atoms happens in 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.

DOI
10.1116/5.0281991

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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 the Thomas–Fermi approximation and, for an isotropic confinement, we find that the highest accumulation of atoms happens in 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.
This thesis investigates the geometric properties of dipolar Bose-Einstein condensates confined in harmonic and bubble-shaped traps.We develop a set of analytical and variational methods to describe the geometric shape of the condensate, the corresponding dipole-induced deformation, as well as the transition between filled and hollow clouds.Within the Thomas-Fermi regime in the Gross-Pitaevskii mean field theory, we present a convolution-based approximation and a generalized analytic solution to the Thomas-Fermi equation in the context of polynomial trapping potentials.Beyond mean-field, we explore the emergence of angular density modulations driven by quantum fluctuations and provide a variational model to estimate the number of droplets forming in the bubble trap.
This thesis investigates the geometric properties of dipolar Bose-Einstein condensates confined in harmonic and bubble-shaped traps. We develop a set of analytical and variational methods to describe the geometric shape of the condensate, the corresponding dipole-induced deformation, as well as the transition between filled and hollow clouds. Within the Thomas-Fermi regime in the Gross-Pitaevskii mean field theory, we present a convolution-based approximation and a generalized analytic solution to the Thomas- Fermi equation in the context of polynomial trapping potentials. Beyond mean-field, we explore the emergence of angular density modulations driven by quantum fluctuations and provide a variational model to estimate the number of droplets forming in the bubble trap.
We analyze the the remarkable analogy between the classical Klein–Gordon equation for a test scalar field in a flat and also in a curved background, and the Gross–Pitaevskii equation for a Bose–Einstein condensate trapped by an external potential. We stress here that the solution associated with the Klein–Gordon equation (KG) in a flat space time has the same mathematical structure, under certain circumstances, to those obtained for the Gross–Pitaevskii equation, that is, a static soliton solution. Additionally, Thomas–Fermi approximation is applied to the 3–dimensional version of this equation, in order to calculate some thermodynamical properties of the system in curved a space–time back ground. Finally, we stress the fact that a gravitational background provides, in some cases, a kind of confining potential for the scalar field, allowing us to remarks even more the possible connection between scalar fields and the phenomenon of Bose–Einstein condensation.
We study the interplay between the anisotropy of the dipole-dipole interaction and confinement in a curved geometry by means of the extended Gross-Pitaevskii equation, which allows us to characterize the ground state of a dipolar Bose gas under the confinement of a bubble trapping potential. We do so in terms of the scattering length a and the number of particles. We observe the emergence of a wide variety of dipolar solids, consisting on arrangements of different number of droplets along a ring over the equator of the spherical shell confinement. We also show that the transition between the different phases of the system can be engineered by varying a , the number of particles or the radius of the trap, parameters, which can be experimentally tuned. Finally, we show the importance of working in microgravity conditions as gravity unstabilizes the observed dipolar solids. Published by the American Physical Society 2024
Bose–Einstein condensates (BECs) comprise a state of matter in which separate atoms or subatomic particles coalesce into a single macroscopic wave function when cooled to near absolute zero. BECs are confined within traps when carrying out experiments, and in the present paper, we model BECs confined within three-dimensional infinite potential wells corresponding to impenetrable boxes with prescribed geometric properties. To do this, we study the Gross–Pitaevskii equation (GPE) governing the macroscopic wave function within generic three-dimensional finite boxes, first describing analytical and numerical approaches for solving the ground state macroscopic wave function, and then employing these methods to understand how the geometry of a confining box influences properties of the BEC wave function. We show that the chemical potential of the BEC is sensitive to the size and shape of the box, with the chemical potential lowest for a spherical box and comparatively large for boxes with more extreme aspect ratios. Compared to commonly employed harmonic traps, geometrically confined BECs exhibit larger variations of the chemical potential with the self-interaction parameter, yet geometrically confined attractive BECs may be more robust against collapse. In particular, we show that the collapse threshold for an attractive BEC can be modified by the specific manner of geometric confinement.
Ground states and dynamical properties of a dipolar Bose–Einstein condensate are analyzed based on the Gross–Pitaevskii–Poisson system (GPPS) and its dimension reduction models under an anisotropic confining potential. We begin with the three-dimensional (3D) GPPS and review its quasi-two-dimensional (2D) approximate equations when the trap is strongly confined in the z-direction and quasi-one-dimensional (1D) approximate equations when the trap is strongly confined in the x- and y-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator (−Δ)1/2(-\Delta)^{1/2}(−Δ)1/2 is involved which brings significant difficulties into the analysis. Existence and uniqueness as well as nonexistence of the ground state under different parameter regimes are established for the quasi-2D and quasi-1D equations. Well-posedness of the Cauchy problem for both types of equations and finite time blow-up in two dimensions are analyzed. Finally, we rigorously prove the convergence with linear convergence rate for the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in the weak interaction regime.