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R. Labouvie et al. [Phys. Rev. Lett. 116, 235302 (2016)] have described anexperiment with a weakly interacting Bose-Einstein condensate trapped in aone-dimensional optical lattice with localized loss created by a focusedelectron beam. We show that by setting suitable initial currents betweenneighboring sites it is possible to create $\mathcal{PT}$-symmetricquasi-stationary and $\mathcal{PT}$-symmetry broken decaying states in anembedded two-mode subsystem. This subsystem exhibits gain provided by thecoupling to the reservoir sites and localized loss due to the electron beam,and shows the same dynamics as a non-Hermitian two-mode system with symmetricreal and antisymmetric imaginary time-independent potentials, except for aproportionality factor in the chemical potential. We also show that there arethree other equivalent scenarios, and that the presence of a localized lossterm significantly reduces the size of the condensate required for therealization.

Realization of $\mathcal{PT}$-symmetric and $\mathcal{PT}$-symmetry broken states in static optical lattice potentials