Radially Symmetric Nonlinear States of Harmonically Trapped Bose-Einstein Condensates

TL;DR

This paper systematically constructs and analyzes the stability of radially symmetric nonlinear states in two-dimensional Bose-Einstein condensates within harmonic traps, revealing stability conditions for ground and excited states under various interactions and topological charges.

Contribution

It introduces a comprehensive set of nonlinear solutions based on the quantum harmonic oscillator spectrum and analyzes their stability, including effects of vorticity and interaction type.

Findings

Ground state is linearly stable for repulsive interactions.

Higher excited states are generally unstable.

Stability of the ground state depends on interaction type and atom number.

Abstract

Starting from the spectrum of the radially symmetric quantum harmonic oscillator in two dimensions, we create a large set of nonlinear solutions. The relevant three principal branches, with $n_r=0,1$ and 2 radial nodes respectively, are systematically continued as a function of the chemical potential and their linear stability is analyzed in detail, in the absence as well as in the presence of topological charge $m$, i.e., vorticity. It is found that for repulsive interatomic interactions {\it only} the ground state is {\it linearly stable} throughout the parameter range examined. Furthermore, this is true for topological charges $m=0$ or $m=1$; solutions with higher topological charge can be unstable even in that case. All higher excited states are found to be unstable in a wide parametric regime. However, for the focusing/attractive case the ground state with $n_r=0$ and $m=0$ can only be stable for a sufficiently low number of atoms. Once again, excited states are found to be generically unstable. For unstable profiles, the dynamical evolution of the corresponding branches is also followed to monitor the temporal development of the instability.