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Institute of Physics eBooks

Bose–Einstein condensates with long-range interactions

Paulsamy Muruganandam, Ramaswamy Radha

Published Nov 1, 2025• 0 citations
Publisher

Abstract

Chapter 8 is devoted to the analysis of dipolar Bose–Einstein condensates with long-range anisotropic tunable dipole–dipole interactions, using mean-field Gross–Pitaevskii equations in one-, two-, and three-dimensional traps. Split-step Crank–Nicolson method, Gaussian variational approach, and Thomas–Fermi approximations are applied to study ground-state properties, nonlinear excitations, and stability across trap geometries.

DOI
10.1088/978-0-7503-5447-9ch8

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Chapter 8 is devoted to the analysis of dipolar Bose–Einstein condensates with long-range anisotropic tunable dipole–dipole interactions, using mean-field Gross–Pitaevskii equations in one-, two-, and three-dimensional traps. Split-step Crank–Nicolson method, Gaussian variational approach, and Thomas–Fermi approximations are applied to study ground-state properties, nonlinear excitations, and stability across trap geometries.
Bose-Einstein condensates with an attractive 1/r1/r1/r interaction and with dipole-dipole interaction are investigated in the framework of the Gaussian variational ansatz introduced by S. Rau, J. Main, and G. Wunner [Phys. Rev. A 82, 023610 (2010)]. We demonstrate that the method of coupled Gaussian wave packets is a full-fledged alternative to direct numerical solutions of the Gross-Pitaevskii equation, or even superior in that coupled Gaussians are capable of producing both stable and unstable states of the Gross-Pitaevskii equation and thus of giving access to yet unexplored regions of the space of solutions of the Gross-Pitaevskii equation. As an alternative to numerical solutions of the Bogoliubov--de Gennes equations, the stability of the stationary condensate wave functions is investigated by analyzing the stability properties of the dynamical equations of motion for the Gaussian variational parameters in the local vicinity of the stationary fixed points. For blood-cell-shaped dipolar condensates it is shown that on the route to collapse the condensate passes through a pitchfork bifurcation, where the ground state itself turns unstable, before it finally vanishes in a tangent bifurcation.
The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.
The variational method of coupled Gaussian functions is applied to Bose-Einstein condensates with long-range interactions. The time dependence of the condensate is described by dynamical equations for the variational parameters. We present the method and analytically derive the dynamical equations from the time-dependent Gross-Pitaevskii equation. The stability of the solutions is investigated using methods of nonlinear dynamics. The concept presented in this article will be applied to Bose-Einstein condensates with monopolar 1/r1/r1/r and dipolar 1/r31/{r}^{3}1/r3 interaction in the subsequent article [S. Rau et al., Phys. Rev. A 82, 023611 (2010)], where we will present a wealth of phenomena obtained using the ansatz with coupled Gaussian functions.
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.
We study properties of a trapped dipolar Bose-Einstein condensate (BEC) in a circular ring or a spherical shell using the mean-field Gross-Pitaevskii equation. In the case of the ring-shaped trap we consider different orientations of the ring with respect to the polarization direction of the dipoles. In the presence of long-range anisotropic dipolar and short-range contact interactions, the anisotropic density distribution of the dipolar BEC in both traps is discussed in detail. The stability condition of the dipolar BEC in both traps is illustrated in a phase plot of dipolar and contact interactions. We also study and discuss the properties of a vortex dipolar BEC in these traps.