# Using impure many-body particle states to generate exact $\mathcal{PT}$-symmetry in a time-dependent four-well system. (arXiv:1802.01323v1 [quant-ph])

Bose-Einstein condensates with balanced gain and loss in a double-well
potential have been shown to exhibit PT-symmetric states. As proposed by
Kreibich et al [Phys. Rev. A 87, 051601(R) (2013)], in the mean-field limit the
dynamical behaviour of this system, especially that of the PT-symmetric states,
can be simulated by embedding it into a four-well system with time-dependent
parameters. In this paper we go beyond the mean-field approximation and
investigate many-body effects in this system, which are in lowest order
described by the single-particle density matrix. The conditions for PT symmetry
in the single-particle density matrix cannot be completely fulfilled by using
pure initial states. Here we show that it is mathematically possible to achieve
exact PT symmetry in the four-well many-body system in the sense of the
dynamical behaviour of the single-particle density matrix. In contrast to
previous work, for this purpose, we use impure initial states fulfilling
certain constraints and use them to calculate the dynamics.