Casimir-Polder effect for a stack of conductive planes

The Casimir-Polder interaction between an atom and a multilayered system composed of infinitely thin planes is considered using the $\zeta$-function regularization approach with zero-point energies summation. As a prototype material, each plane is represented by a graphene sheet, an atomically thin layer of carbon atoms organized in a hexagonal lattice, whose optical response is described by a constant conductivity or Drude-Lorentz model conductivity. Asymptotic expressions for various separations are derived and compared to numerical calculations. We distinguish between large atom-plane distance limit, where retardation effects are prominent, and small atom-plane distance limit, where the typical van der Waals coefficient is found to be dependent on the number of graphenes and characteristic separations. The calculated energies for different atoms and graphene conductivity models brings forward the basic science of the Casimir-Polder effect and suggests ways to manipulate this interaction experimentally.

Figure 1

The $\mathcal{N}$ parallel planes are located at points $z=a,a+d,a+2d,a+3d,...,a+(\mathcal{N}-1)d$. The half space $z\le 0$ is filled by dielectric media with permeability $\varepsilon \left(\omega \right)$.

Figure 2

Left: the function $q\left(x\right)$ as a function of $x$. Right : the ratio $E^{\left(\mathcal{N}\right)}/E_{a\to 0}^{\left(\mathcal{N}\right)}$ for hydrogen atom and for different number of planes $\mathcal{N}=1,2,\infty$ (from bottom to top). We adopt the constant conductivity model with $\eta ={\eta }_{gr}$ and interplane distance $d=0.3354\text{nm}$.

Figure 3

The Casimir-Polder energy of a hydrogen atom, normalized by the Casimir-Polder energy atom-ideal metal, near a stack of graphene planes with conductivity described using DL model for $a={10}^n,n=0,1,2,3,\infty$. The interplane graphene separation is the equilibrium distance for graphite $d=0.3345\text{nm}$.

Figure 4

The Casimir-Polder energy multiplied by $a^4$ for several atoms as a function of number of graphene planes $\mathcal{N}$. The conductivity for each graphene plane is taken into account using the DL model and the interplane distance is $d=0.3345\text{nm}$ while $a=100\text{nm}$.

Figure 5

The Casimir-Polder energy for hydrogen (solid lines) and Rb (dashed lines) atom near a stack of graphene planes with DL (top two curves) and ${\eta }_{gr}$ (bottom two curves) conductivities. Here $d=0.3345\text{nm}$ and $a=100\text{nm}$.

Figure 6

The vdW coefficient $C_3(a,\mathcal{N})$ for several light atoms and for two models of the conductivity. Solid lines: $\mathcal{N}=1$ and dashed lines: $\mathcal{N}=\infty$. The insert is the relative difference given by Eq. (11) as a function of $a$ separation.

Figure 7

The vdW coefficient $C_3(a,\mathcal{N})$ for several heavy atoms and for two models of the conductivity. Solid lines: $\mathcal{N}=1$ and dashed lines: $\mathcal{N}=\infty$. The insert is relative the difference given by Eq. (11) as a function of separation $a$.