Bosonic Josephson effect in the Fano-Anderson model

We investigate the coherent dynamics of a non-interacting Bose-Einstein condensate in a system consisting of two bosonic reservoirs coupled via a spatially localized mode. We describe this system by a two-terminal Fano-Anderson model and investigate analytically the time evolution of observables such as the bosonic Josephson current. In doing so, we find that the Josephson current sensitively depends on the on-site energy of the localized mode. This facilitates to use this setup as a transistor for a Bose-Einstein condensate. We identify two regimes. In one regime, the system exhibits well-behaved long-time dynamics with a slowly oscillating and undamped Josephson current. In a second regime, the Josephson current is a superposition of an extremely weakly damped slow oscillation and an undamped fast oscillation. Our results are confirmed by finite-size simulations.