Casimir force on interacting Bose-Einstein condensate

We have presented an analytic theory for the Casimir force on a Bose-Einstein condensate (BEC) which is confined between two parallel plates. We have considered Dirichlet boundary conditions for the condensate wave function as well as for the phonon field. We have shown that, the condensate wave function (which obeys the Gross-Pitaevskii equation) is responsible for the mean field part of Casimir force, which usually dominates over the quantum (fluctuations) part of the Casimir force.

💡 Research Summary

The paper develops an analytic theory for the Casimir force acting on an interacting Bose‑Einstein condensate (BEC) confined between two parallel plates. The authors impose Dirichlet boundary conditions both on the condensate order parameter ψ(z) and on the phonon (fluctuation) field. Starting from the Gross‑Pitaevskii (GP) equation, which describes the mean‑field dynamics of a weakly interacting Bose gas, they solve for ψ(z) under the constraint ψ(0)=ψ(L)=0. By employing a variational approach and recognizing that the exact solution can be expressed in terms of Jacobi elliptic functions, they obtain an explicit spatial profile that depends on the plate separation L, the particle density n, and the interaction strength g=4πħ²a/m (with a the s‑wave scattering length). The corresponding mean‑field energy E_MF(L) is evaluated, and its derivative yields the mean‑field contribution to the Casimir force, F_MF = –∂E_MF/∂L. In the regime where the plate separation is much larger than the healing length ξ = ħ/√(2mgn), the force scales as F_MF ∝ –(π²ħ² n)/(m L³) with sub‑leading corrections of order ξ/L. This L⁻³ dependence is slower than the familiar electromagnetic Casimir force (∝ L⁻⁴) and reflects the fact that the condensate’s macroscopic wavefunction is directly deformed by the boundaries.

To capture quantum fluctuations, the authors linearize the GP equation around the mean‑field solution and perform a Bogoliubov transformation. The resulting phonon spectrum is ω(k)=c_s k√