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

摘要

The most important properties of a Bose-Einstein condensate subject tobalanced gain and loss can be modelled by a Gross-Pitaevskii equation with anexternal $\mathcal{PT}$-symmetric double-delta potential. We study its linearvariant with a supersymmetric extension. It is shown that both in the$\mathcal{PT}$-symmetric as well as in the $\mathcal{PT}$-broken phasearbitrary stationary states can be removed in a supersymmetric partnerpotential without changing the energy eigenvalues of the other state. Thecharacteristic structure of the singular delta potential in the supersymmetryformalism is discussed, and the applicability of the formalism to the nonlinearGross-Pitaevskii equation is analysed. In the latter case the formalism couldbe used to remove $\mathcal{PT}$-broken states introducing an instability tothe stationary $\mathcal{PT}$-symmetric states.