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arXiv:2106.15736 · Physical Review A

Hyperspherical approach to dipolar droplets beyond the mean-field limit

Eli Halperin, John L. Bohn

Published Sep 29, 2021• 1 citation
arXivHTMLPDF

Abstract

We apply a hyperspherical formulation to a trapped Bose-Einstein condensate with dipolar and contact interactions. Central to this approach is a general correspondence between $K$-harmonic hyperspherical methods and a suitable Gaussian ansatz to the Gross-Pitaevskii equation, regardless of the form of the interparticle potential. This correspondence allows one to obtain hyperspherical potential energies for a wide variety of physical problems. In the case of the dipolar Bose-Einstein condensate, this motivates the inclusion of a beyond-mean-field term within the hyperspherical picture, which allows us to describe the energies and wave functions of excitations of self-bound dipolar droplets outside the mean-field limit.

DOI
10.1103/physreva.104.033324

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We apply a hyperspherical formulation to a trapped Bose-Einstein condensate with dipolar and contact interactions. Central to this approach is a general correspondence between KKK-harmonic hyperspherical methods and a suitable Gaussian ansatz to the Gross-Pitaevskii equation, regardless of the form of the interparticle potential. This correspondence allows one to obtain hyperspherical potential energies for a wide variety of physical problems. In the case of the dipolar Bose-Einstein condensate, this motivates the inclusion of a beyond-mean-field term within the hyperspherical picture, which allows us to describe the energies and wave functions of excitations of self-bound dipolar droplets outside the mean-field limit.
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We study a system of AAA identical interacting bosons trapped by an external field by solving ab initio the many-body Schr\"odinger equation. A complete solution by using, for example, the traditional hyperspherical harmonics (HH) basis develops serious practical problems due to the large degeneracy of HH basis. Symmetrization of the wave function, calculation of the matrix elements, etc., become an immensely formidable task as AAA increases. Instead of the HH basis, here we use a new basis, called ``potential harmonics'' (PH) basis, which is a subset of HH basis. We assume that the contribution to the orbital and grand orbital [in $3(A\ensuremath{-}1)$-dimensional space of the reduced motion] quantum numbers comes only from the interacting pair. This implies inclusion of two-body correlations only and disregard of all higher-body correlations. Such an assumption is ideally suited for the Bose-Einstein condensate (BEC), which is required, for experimental realization of BEC, to be extremely dilute. Hence three and higher-body collisions are almost totally absent. Unlike the $(3A\ensuremath{-}4)$ hyperspherical variables in HH basis, the PH basis involves only three active variables, corresponding to three quantum numbers---the orbital lll, azimuthal mmm, and the grand orbital 2K+l2K+l2K+l quantum numbers for any arbitrary AAA. It drastically reduces the number of coupled equations and calculation of the potential matrix becomes tremendously simplified, as it involves integrals over only three variables for any AAA. One can easily incorporate realistic atom-atom interactions in a straightforward manner. We study the ground and excited state properties of the condensate for both attractive and repulsive interactions for various particle number. The ground state properties are compared with those calculated from the Gross-Pitaevskii equation. We notice that our many-body results converge towards the mean field results as the particle number increases.
Abstract We study the beyond-mean-field corrections to the energy of a dipolar Bose gas confined to two dimensions by a box potential with dipoles oriented in plane such that their interaction is anisotropic in the two unconfined dimensions. At a critical strength of the dipolar interaction the system becomes unstable on the mean field level. We find that the ground state of the gas is strongly influenced by the corrections, leading to formation of a self-bound droplet, in analogy to the free space case. Properties of the droplet state can be found by minimizing the extended Gross–Pitaevskii energy functional. In the limit of strong confinement we show analytically that the correction can be interpreted as an effective three-body repulsion which stabilizes the gas at finite density.
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.
A variational basis set motivated by mean-field theory is utilized to describe the Bose-Einstein condensate within the adiabatic hyperspherical coordinate framework. The simplest single-orbital variant of this treatment reproduces many of the ground-state properties predicted by the Gross-Pitaevskii equation. But a multiorbital improvement to the basis set yields a better representation of particle correlations and of the critical number where the condensate collapses for a negative two-body scattering length. The method also produces systematic deviations from Bogoliubov theory for the fundamental monopole excitation frequency.