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Mathematical methods for geometric characterization of dipolar Bose-Einstein condensates: bridging harmonic and shell traps

Pedro de Castro Diniz

Published May 21, 2025• 0 citations
Publisher

Abstract

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.

DOI
10.11606/t.76.2025.tde-18072025-081212

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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 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
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.
We develop theory for a flattened dipolar Bose–Einstein condensate produced by harmonic confinement along one direction. The role of both short-ranged contact interactions and long-ranged dipole–dipole interactions is considered, and the dipoles are allowed to be polarized along an arbitrary direction. We discuss the symmetry properties of the condensate and the part of the excitation spectrum determining stability, and introduce two effective interaction parameters that allow us to provide a general description of the condensate properties, rotons, and stability. We diagonalize the full theory to obtain benchmark results for the condensate and quasiparticle excitations, and characterize the exact mean field stability of the system. We provide a unified formulation for a number of approximate schemes to describe the condensate and quasiparticles, including the standard quasi-two-dimensional approximation, two kinds of variational ansatz, and a Thomas–Fermi approximation. Some of these schemes have been widely used in the literature despite not being substantiated against the exact theory. We provide this validation and establish the regimes where the various theories perform well.
The recent developments of microgravity experiments with ultracold atoms have produced a relevant boost in the study of shell-shaped ellipsoidal Bose-Einstein condensates. For realistic bubble-trap parameters, here we calculate the critical temperature of Bose-Einstein condensation, which, if compared to the one of the bare harmonic trap with the same frequencies, shows a strong reduction. We simulate the zero-temperature density distribution with the Gross-Pitaevskii equation, and we study the free expansion of the hollow condensate. While part of the atoms expands in the outward direction, the condensate self-interferes inside the bubble trap, filling the hole in experimentally observable times. For a mesoscopic number of particles in a strongly interacting regime, for which more refined approaches are needed, we employ quantum Monte Carlo simulations, proving that the nontrivial topology of a thin shell allows superfluidity. Our work constitutes a reliable benchmark for the forthcoming scientific investigations with bubble traps.