Quantum effective action for the bosonic Josephson junction

We investigate a bosonic Josephson junction by using the path-integral formalism with relative phase and population imbalance as dynamical variables. We derive an effective only-phase action performing functional integration over the population imbalance. We then analyze the quantum effective only-phase action, which formally contains all the quantum corrections. To the second order in the derivative expansion and to the lowest order in $\hbar$, we obtain the quantum correction to the Josephson frequency of oscillation. Finally, the same quantum correction is found by adopting an alternative approach. Our predictions are a useful theoretical tool for experiments with atomic or superconducting Josephson junctions.