Multiple Scattering Expansion for Dielectric Media: Casimir Effect

Recent measurements of Casimir forces have provided evidence of an intricate modification of quantum fluctuations of the electromagnetic field in complex geometries. Here we introduce a multiple scattering description for Casimir interactions between bodies of arbitrary shape and material composition, admitting an expansion as a sequence of inter- and intra-body wave scatterings. Interactions in complex geometries can be computed within the current experimental resolution from typically a few wave scatterings, notably without any a priori knowledge of the scattering amplitudes of the bodies. Some first applications demonstrate the power of the approach.

Figure 1

Diagrammatic representation of contributions to the MSE, shown in panel (a) for the scattering Green’s function $ℾ(\mathbf{r},{\mathbf{r}}^{\prime })$ of a single body with source point ${\mathbf{r}}^{\prime }$ and observation point $\mathbf{r}$, and in panel (b) for the Casimir energy between two bodies. In the displayed examples, lines with arrows represent free propagation between surface points of the same body (blue lines) and to external points or between surface points of different bodies (magenta lines). See text for details.

Figure 2

Different orders of the MSE for the Casimir energy between a plate made of doped silicon and a plate made of gold (see Supplemental Material for details), normalized to the known exact energy. Indices of ${\mathrm{MSE}}_{kl}$ label the number of scatterings between the plates [$2(k+1)$] and within the silicon plate ($l$) (see text for details).

Figure 3

(a) Schematic of the wedge-plate configuration with edge-plate separation $d$ and angle $\theta$ between the plate and the sides of the wedge. Other panels (b) to (d) show the Casimir energy of the wedge-plate configuration both at zero temperature and at room temperature ($T=300\mathrm{K}$) as a function of the separation $d$, obtained from the multiple scattering expansion (MSE), and the proximity force approximation (PFA), for three different angles $\theta =\pi /6$, $\theta =\pi /3$, and $\theta =4\pi /9$.

Figure 4

Influence of material properties and thermal fluctuations on the wedge-plate interaction. Modification factor $\chi$ which compares Casimir energy ${\mathcal{E}}_{\mathrm{Au}\text{−}\mathrm{Si}}$ of the Au-Si plate wedge to energy for a perfect metal (PM) plate wedge, both at $T=0\mathrm{K}$ and $T=300\mathrm{K}$. Thin curves with open dots represent ${\chi }_{0K}={\mathcal{E}}_{\mathrm{Au}\text{−}\mathrm{Si},0\mathrm{K}}/{\mathcal{E}}_{\mathrm{PM},0\mathrm{K}}$, and bold curves with filled dots show ${\chi }_{300\mathrm{K}}={\mathcal{E}}_{\mathrm{Au}\text{−}\mathrm{Si},300\mathrm{K}}/{\mathcal{E}}_{\mathrm{PM},300\mathrm{K}}$. See text for details.