Cusp bifurcation in the eigenvalue spectrum of PT-symmetric Bose-Einstein condensates

TL;DR

This paper investigates how balanced gain and loss in a Bose-Einstein condensate alter the bifurcation structure of stationary states, revealing a cusp bifurcation in the eigenvalue spectrum.

Contribution

It introduces a bicomplex formulation to analyze the transition, uncovering three tangent bifurcations coalescing into a cusp bifurcation under small gain-loss effects.

Findings

Identification of three tangent bifurcations in the eigenvalue spectrum.

Discovery of a cusp bifurcation arising from coalescing bifurcations.

Detailed analysis of bifurcation scenario changes due to gain and loss.

Abstract

A Bose-Einstein condensate in a double-well potential features stationary solutions even for attractive contact interaction as long as the particle number and therefore the interaction strength do not exceed a certain limit. Introducing balanced gain and loss into such a system drastically changes the bifurcation scenario at which these states are created. Instead of two tangent bifurcations at which the symmetric and antisymmetric states emerge, one tangent bifurcation between two formerly independent branches arises [D. Haag et al., Phys. Rev. A 89, 023601 (2014)]. We study this transition in detail using a bicomplex formulation of the time-dependent variational principle and find that in fact there are three tangent bifurcations for very small gain-loss contributions which coalesce in a cusp bifurcation.