The Casimir-Lifshitz Anharmonic Oscillator

We analyze the Casimir-Lifshitz effect associated with the electromagnetic field in the presence of a rectangular waveguide consisting of two distinct dielectric materials in a (3 + 1)-dimensional spacetime. We employ the surface mode technique to derive a generalized Lifshitz formula for this specific geometry. Our formulation accounts for the unique dielectric properties of the materials composing the waveguide, leading to a precise calculation of the Casimir-Lifshitz energy. In the asymptotic limit, our results recover the classical expressions for perfect reflecting boundaries. The behavior of the force on the rectangular cavity walls admit us to identify the system as a Casimir-Lifshitz anharmonic oscillator, that may vibrate due only the to vacuum stress if we allow two parallel non-fixed walls.