Casimir Energies in a One-Dimensional Cavity with a Fluctuating Boundary

Abstract

We consider a massless scalar field in a one-dimensional cavity with one fixed and one mobile wall. We assume that the mobile wall is also subjected to a harmonic potential, and its mechanical degrees of freedom are treated quantum-mechanically. The wall's position has thus quantum fluctuations around the equilibrium position. The possible motion of the wall makes the cavity length variable, and this gives rise to a wall-field interaction and an effective interaction between the modes of the cavity. We use an effective Hamiltonian, originally introduced by C. K. Law, to describe our system in terms of field modes relative to the equilibrium position of the mobile wall. We obtain by perturbation theory the dressed ground state of the wall-field coupled system, which contains pairs of virtual quanta of the field and excitations of the wall's mechanical degrees of freedom. We evaluate the average number of virtual excitations in each mode of the cavity, induced by the effective wall-field interaction, as well as the renormalized field energy density inside the cavity. We show that the quantum fluctuations of the wall's position significantly affect the field energy density in the cavity, in particular in the proximity of the mobile wall. We also consider a statistical average of the energy density on the position of the fluctuating boundary, in order to discuss the known problem of the divergence of field energy densities at the boundaries. All these quantities are then compared with analogous quantities for a cavity with fixed walls. We find a correction to the Casimir potential energy and to the field energy density in the cavity, due to the position fluctuations of the cavity wall, and discuss how these corrections depend on relevant parameters of the mobile wall, in particular its plasma frequency, mass and frequency of the harmonic potential. Observability of these new effects is also discussed, as well as the relation to the dynamical Casimir effect.