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Métodos matemáticos para caracterização geométrica de condensados de Bose-Einstein dipolares: conectando armadilhas harmônicas e de casca

Pedro Paulo Vissotto de Paiva Diniz

Published May 21, 2025• 0 citations
OpenAlex

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.

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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 derive rigorous one- and two-dimensional mean-field equations for cigar- and pancake-shaped dipolar Bose-Einstein condensates with arbitrary polarization angle. We show how the dipolar interaction modifies the contact interaction of the strongly confined atoms. In addition, our equations introduce a nonlocal potential, which is anisotropic for pancake-shaped condensates. We propose to observe this anisotropy via measurement of the condensate aspect ratio. We also derive analytically approximate density profiles from our equations. Both the numerical solutions of our reduced mean-field equations and the analytical density profiles agree well with numerical solutions of the full Gross-Pitaevskii equation while being more efficient to compute.
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 investigate the formation, stability, and dynamics of three-dimensional ring-shaped and multipole vortical quantum droplets (QDs) in nonrotating dipolar Bose-Einstein condensates held in a toroidal trapping potential. The QD dynamics are investigated in the framework of the extended Gross-Pitaevskii equation, which includes long-range dipole-dipole interactions (DDIs) and the beyond-mean-field Lee-Huang-Yang (LHY) term, revealing the emergence of self-bound states. Stable stationary solutions for multipole QDs with different values of the topological charge (vorticity S) are shaped as necklacelike modes, with the number of "beads" (multipole order) n=2S, up to S=6. The stability area of the multipoles shrinks with the increase of S. For higher values of S the centrifugal effect associated with the phase winding destabilizes the ring-shaped QDs and drives the formation of fragmented multipole droplet states. The dependence of the chemical potential, total energy, and peak density on the norm (number of particles) and S is produced. These findings uncover the stabilizing effect of the LHY correction and DDI anisotropy in maintaining complex QD states in the nonrotating configurations.