Casimir pressure with dissipation and quantum corrections from axion dark matter

We study the effects of the oscillating axion field present in our environment on the Casimir pressure between two metallic plates. We take into account the finite conductivity of the boundary plates and model the interactions between matter and photons in the Schwinger-Keldysh formalism. This allows us to take into account dissipation in the quantum field description of this open quantum system and retrieve the Lifschitz results for the Casimir interaction between two metallic plates. We then compute the leading correction to the Lifschitz theory in inverse powers of the axion suppression scale and show that the Casimir pressure receives oscillating corrections depending on the product of the axion mass and the distance between the plates. This contribution is repulsive at large distance compared to the axion Compton wavelength as a consequence of the breaking of parity invariance by the axion dark matter background.