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Bose-Einstein condensates with balanced gain and loss in a double-wellpotential have been shown to exhibit PT-symmetric states. As proposed byKreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit thedynamical behaviour of this system, especially that of the PT-symmetric states,can be simulated by embedding it into a Hermitian four-well system withtime-dependent parameters. In this paper we go beyond the mean-fieldapproximation and investigate many-body effects in this system, which are inlowest order described by the single-particle density matrix. The conditionsfor PT symmetry in the single-particle density matrix cannot be completelyfulfilled by using pure initial states. Here we show that it is mathematicallypossible to achieve exact PT symmetry in the four-well many-body system in thesense of the dynamical behaviour of the single-particle density matrix. Incontrast to previous work, for this purpose, we use mixed initial statesfulfilling certain constraints and use them to calculate the dynamics.

Using mixed many-body particle states to generate exact $\mathcal{PT}$-symmetry in a time-dependent four-well system