Casimir Energy of a BEC: From Moderate Interactions to the Ideal Gas

Considering the Casimir effect due to phononic excitations of a weakly interacting dilute {BEC}, we derive a re-normalized expression for the zero temperature Casimir energy $\mathcal{E}_c$ of a {BEC} confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to the free energy of a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, $\mathcal{E}_c$ smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal {BEC}. For the weakly interacting case, $\mathcal{E}_c$ reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.