Casimir force inadequacy in explaining a strong attractive force in a micrometer-sized narrow-gap re-entrant cavity

TL;DR

The paper interrogates whether the Casimir force can explain the strong attraction observed in a micrometer-scale narrow-gap re-entrant cavity coupled to a SiN membrane. The authors compute the Casimir spring constant $k_C(x)$ using the Proximity Force Approximation (PFA) within Lifshitz theory, modeling the Al post and the Au/Nb-coated membrane as thick planar slabs. They find that $k_C$ is orders of magnitude smaller than the membrane spring constant $k_S$ (approximately $k_S \approx 572\,\mathrm{N/m}$ for Au and $k_S \approx 949\,\mathrm{N/m}$ for Nb), so Casimir forces cannot account for the observed increase in effective stiffness $k_{ m eff}$ with gap, which scales roughly as $x^{-4}$, while the force itself scales as $F_C \sim x^{-3}$. The results point to alternative explanations, such as electrostatic forces arising from surface potential variations, and suggest Kelvin probe measurements to test this possibility.

Abstract

Pate et al. \cite{pate} investigated a macroscopic opto-mechanical system with a narrow-gap re-entrant cavity coupled to a SiN membrane resonator coated with Au or Nb. They observed a significant increase in the membrane's effective spring constant $k_{\rm eff}$ for sub-2-micron gaps $x$. This increase scales roughly with $x^{-4}$, suggesting an attractive force pulling the membrane towards the re-entrant Al post, with an $x^{-3}$ dependence. Attributing this force solely to the thermal Casimir effect is challenged by our detailed calculations (presented below). These calculations reveal that the Casimir force, at the investigated gap sizes, is orders of magnitude weaker than the observed force. This significant discrepancy necessitates an alternative explanation for the observed attraction.

Figures (1)

• Figure 1: Casimir spring constant $k_{\rm C}=F'_{\rm C}(x)$ as a function of gap size, $x$. The spring constant is shown for an Au-coated membrane (blue solid line) a Nb-coated membrane (red dashed line), both computed for room temperature $T=300$ K. The green dotted line shows the Casimir spring constant $k^{\rm (pc)}_{\rm C}$ in the limit of a perfectly conducting cavity at $T=0$ (see Eq. (\ref{['pc']})).