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We present a quantum master equation describing a Bose-Einstein condensate with particle loss on one lattice site and particle gain on the other lattice site whose mean-field limit is a non-Hermitian PT-symmetric Gross-Pitaevskii equation. It is shown that the characteristic properties of PT-symmetric systems, such as the existence of stationary states and the phase shift of pulses between two lattice sites, are also found in the many-particle system. Visualizing the dynamics on a Bloch sphere allows us to compare the complete dynamics of the master equation with that of the Gross-Pitaevskii equation. We find that even for a relatively small number of particles the dynamics are in excellent agreement and the master equation with balanced gain and loss is indeed an appropriate many-particle description of a PT-symmetric Bose-Einstein condensate.
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In this work we present a new generic feature of PT-symmetric Bose-Einstein condensates by studying the many-particle description of a two-mode condensate with balanced gain and loss. This is achieved using a master equation in Lindblad form whose mean-field limit is a PT-symmetric Gross-Pitaevskii equation. It is shown that the purity of the condensate periodically drops to small values but then is nearly completely restored. This has a direct impact on the average contrast in interference experiments which cannot be covered by the mean-field approximation, in which a completely pure condensate is assumed.
Most of the work done in the field of Bose-Einstein condensates with balanced gain and loss has been performed in the mean-field approximation using the PT-symmetric Gross-Pitaevskii equation. In this work we study the many-particle dynamics of a two-mode condensate with balanced gain and loss described by a master equation in Lindblad form whose purity periodically drops to small values but then is nearly completely restored. This effect cannot be covered by the mean-field approximation, in which a completely pure condensate is assumed. We present analytic solutions for the dynamics in the non-interacting limit and use the Bogoliubov backreaction method to discuss the influence of the on-site interaction. Our main result is that the strength of the purity revivals is almost exclusively determined by the strength of the gain and loss and is independent of the amount of particles in the system and the interaction strength. For larger particle numbers, however, strong revivals are shifted towards longer times, but by increasing the interaction strength these strong revivals again occur earlier.
Balanced gain and loss renders the mean-field description of Bose-Einstein condensates PT symmetric. However, any experimental realization has to deal with unbalancing in the gain and loss contributions breaking the PT symmetry. We will show that such an asymmetry does not necessarily lead to a system without a stable mean-field ground state. Indeed, by exploiting the nonlinear properties of the condensate, a small asymmetry can stabilize the system even further due to a self-regulation of the particle number.
Bose-Einstein condensates with balanced gain and loss can support stationary states despite the exchange of particles with the environment. In the mean-field approximation this is described by the PT-symmetric Gross-Pitaevskii equation with real eigenvalues. In this work we study the role of stationary states in the appropriate many-particle description. It is shown that without particle interaction there exist two non-oscillating trajectories which can be interpreted as the many-particle equivalent of the stationary PT-symmetric mean-field states. Furthermore the system has a non-equilibrium steady state which acts as an attractor in the oscillating regime. This steady state is a pure condensate for strong gain and loss contributions if the interaction between the particles is sufficiently weak.
We investigate the Su-Schrieffer-Heeger model in presence of an injection and removal of particles, introduced via a master equation in Lindblad form. It is shown that the dynamics of the density matrix follows the predictions of calculations in which the gain and loss are modeled by complex $mathcal{PT}$-symmetric potentials. In particular it is found that there is a clear distinction in the dynamics between the topologically different cases known from the stationary eigenstates.
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