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

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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 Ec 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, Ec smoothly describes the transition from the weakly interacting Bogoliubov regime to the non-interacting ideal BEC. For the weakly interacting case, Ec 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.