Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement

Abstract

Ground states and dynamical properties of dipolar Bose-Einstein condensate are analyzed based on the Gross-Pitaevskii-Poisson system (GPPS) and its dimension reduction models under anisotropic confining potential. We begin with the three-dimensional (3D) Gross-Pitaevskii-Poisson system and review its quasi-2D approximate equations when the trap is strongly confined in $z$-direction and quasi-1D approximate equations when the trap is strongly confined in $x$-, $y$-directions. In fact, in the quasi-2D equations, a fractional Poisson equation with the operator $(-\Delta {)}^{1/2}$ 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 equations and finite time blowup in 2D are analyzed. Finally, we rigorously prove the convergence and linear convergence rate between the solutions of the 3D GPPS and its quasi-2D and quasi-1D approximate equations in weak interaction regime.

arXiv comment29 pages

References in corpus (9)

• Bose-Einstein condensation of chromium
• A Strongly Dipolar Bose-Einstein Condensate of Dysprosium
• Numerical Solution of the Gross-Pitaevskii Equation for Bose-Einstein Condensation
• Trapped atomic condensates with anisotropic interactions
• Trapped condensates of atoms with dipole interactions
• Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates
• Novel vortex structures in dipolar condensates
• Convergence rate of dimension reduction in Bose-Einstein condensates
• Effective dipole-dipole interactions in multilayered dipolar Bose-Einstein condensates