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

R. Labouvie et al. [Phys. Rev. Lett. 116, 235302 (2016)] have described an experiment with a weakly interacting Bose-Einstein condensate trapped in a one-dimensional optical lattice with localized loss created by a focused electron beam. We show that by setting suitable initial currents between neighboring sites it is possible to create $\mathcal{P}\mathcal{T}$-symmetric quasi-stationary and $\mathcal{P}\mathcal{T}$-symmetry broken decaying states in an embedded two-mode subsystem. This subsystem exhibits gain provided by the coupling 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 symmetric real and antisymmetric imaginary time-independent potentials, except for a proportionality factor in the chemical potential. We also show that there are three other equivalent scenarios, and that the presence of a localized loss term significantly reduces the size of the condensate required for the realization.