Bose-Einstein condensates with balanced gain and loss beyond mean-field theory

Abstract

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 non-Hermitian PT-symmetric Gross-Pitaevskii equation. However, the exchange of particles with the environment plays a crucial role in such systems which in general leads to deviations from the mean-field behavior. Thus, it is not clear whether a mean-field approach is appropriate. It is the purpose of this work to formulate and study a many-particle description of a Bose-Einstein condensate with balanced gain and loss. This is achieved by using a quantum master equation describing a double well where the incoupling of particles in one well and the outcoupling from the other are implemented with Lindblad superoperators. The in- and outcoupling rates are adjusted in an appropriate manner such that balanced gain and loss is achieved. It is shown that the mean-field limit of this master equation yields a PT-symmetric Gross-Pitaevskii equation. Furthermore the master equation supports the characteristic dynamical properties of PT-symmetric systems. There are, however, fundamental differences compared with the mean-field description revealing a new generic feature of PT-symmetric Bose-Einstein condensates. It is shown that the purity of the condensate periodically drops to small values but then is nearly completely restored, when the particles oscillate in the double well. Since in the mean-field limit a completely pure condensate is assumed, this effect cannot be covered by the Gross-Pitaevskii equation. These purity oscillations have a direct impact on the average contrast in interference experiments. In particular it is found that the extrema of the purity can be precisely measured since the average contrast at these points is not reduced by an imbalance of the particle distribution. To gain a detailed understanding of the purity oscillations, analytic solutions for the dynamics in the non-interacting limit are presented and the Bogoliubov backreaction method is used to discuss the influence of the on-site interaction. A central result is that the strength of the purity revivals does neither depend on the amount of particles in the system nor the interaction strength, but is almost exclusively determined by the strength of the in- and outcoupling processes. However, the strong revivals are shifted towards longer times for larger particle numbers. Without interaction this would make the purity oscillations unobservable for a realistic particle number, but by adjusting the interaction strength the strong revivals again occur earlier.