Supersymmetric model of a Bose-Einstein condensate in a $\mathcal{PT}$-symmetric double-delta trap

TL;DR

This paper explores a supersymmetric extension of a linear model of a Bose-Einstein condensate in a $\,\mathcal{PT}$-symmetric double-delta potential, analyzing state removal and stability implications for nonlinear cases.

Contribution

It introduces a supersymmetric framework for the linear $\,\mathcal{PT}$-symmetric double-delta potential and discusses its potential application to the nonlinear Gross-Pitaevskii equation.

Findings

Supersymmetry allows removal of arbitrary stationary states without changing others' energies.

The formalism addresses the structure of delta potentials within supersymmetry.

Potential to remove $\,\mathcal{PT}$-broken states and influence stability in nonlinear models.

Abstract

The most important properties of a Bose-Einstein condensate subject to balanced gain and loss can be modelled by a Gross-Pitaevskii equation with an external $\mathcal{PT}$-symmetric double-delta potential. We study its linear variant with a supersymmetric extension. It is shown that both in the $\mathcal{PT}$-symmetric as well as in the $\mathcal{PT}$-broken phase arbitrary stationary states can be removed in a supersymmetric partner potential without changing the energy eigenvalues of the other state. The characteristic structure of the singular delta potential in the supersymmetry formalism is discussed, and the applicability of the formalism to the nonlinear Gross-Pitaevskii equation is analysed. In the latter case the formalism could be used to remove $\mathcal{PT}$-broken states introducing an instability to the stationary $\mathcal{PT}$-symmetric states.