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### Casimir interactions for anisotropic magnetodielectric metamaterials

##### F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni

##### Phys. Rev. A **78**, 032117 – Published 26 September 2008

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#### Abstract

We extend our previous work [Phys. Rev. Lett. **100**, 183602 (2008)] on the generalization of the Casimir-Lifsh*tz theory to treat anisotropic magnetodielectric media, focusing on the forces between metals and magnetodielectric metamaterials and on the possibility of inferring magnetic effects by measurements of these forces. We present results for metamaterials including structures with uniaxial and biaxial magnetodielectric anisotropies, as well as for structures with isolated metallic or dielectric properties that we describe in terms of filling factors and a Maxwell Garnett approximation. The elimination or reduction of Casimir “stiction” by appropriate engineering of metallic-based metamaterials, or the indirect detection of magnetic contributions, appear from the examples considered to be very challenging, as small background Drude contributions to the permittivity act to enhance attraction over repulsion, as does magnetic dissipation. In dielectric-based metamaterials the magnetic properties of polaritonic crystals, for instance, appear to be too weak for repulsion to overcome attraction. We also discuss Casimir-Polder experiments, that might provide another possibility for the detection of magnetic effects.

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- Received 16 July 2008

DOI:https://doi.org/10.1103/PhysRevA.78.032117

©2008 American Physical Society

#### Authors & Affiliations

F. S. S. Rosa, D. A. R. Dalvit, and P. W. Milonni

^{}Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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##### Issue

Vol. 78, Iss. 3 — September 2008

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#### Images

###### Figure 1

Typical setup used throughout this paper to compute the Casimir-Lifsh*tz force between a metal and a metamaterial.Reuse & Permissions

###### Figure 2

An incident plane wave impinging on a uniaxial metamaterial with its optic axis perpendicular to the $z=0$ plane.Reuse & Permissions

###### Figure 3

An incident plane wave impinging on a biaxial metamaterial with orthorhombic symmetry (see text).Reuse & Permissions

###### Figure 4

The ratio $F\u2215{F}_{\mathrm{C}}$ for a gold half-space facing an isotropic, interconnected, and silver-based metamaterial. $F\u2215A$ is the Casimir force per unit area in this setup ${F}_{\mathrm{C}}\u2215A=\hslash c{\pi}^{2}\u2215240{a}^{4}$ is the Casimir force per unit area between two perfect plane conductors and $F<0(F>0)$ corresponds to a repulsive (attractive) force. The frequency scale $\Omega =2\pi c\u2215\Lambda $ is chosen as the silver plasma frequency ${\Omega}_{D}=1.43\times {10}^{16}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$. Parameters are for the metal, ${\Omega}_{1}\u2215\Omega =0.96$, ${\gamma}_{1}\u2215\Omega =0.004$, and for the metamaterial, ${\Omega}_{D}\u2215\Omega =1$, ${\gamma}_{D}\u2215\Omega =0.006$, ${\Omega}_{e}\u2215\Omega =0.04$, ${\Omega}_{m}\u2215\Omega =0.1$, ${\omega}_{e}\u2215\Omega ={\omega}_{m}\u2215\Omega =0.1$, ${\gamma}_{e}\u2215\Omega ={\gamma}_{m}\u2215\Omega =0.005$. The inset shows the magnetic permeability ${\mu}_{2}\left(i\xi \right)$ and the electric permittivity ${\u03f5}_{2}\left(i\xi \right)$ of the MM for the different filling factors.Reuse & Permissions

###### Figure 5

The effects of uniaxial anisotropy in the Casimir force between a gold semispace and a metallic-based connected MM with weak Drude background. The distance is fixed to $d=\Lambda $ and repulsion corresponds to negative values of $F\u2215{F}_{\mathrm{C}}$. All parameters are the same as in Fig. 4 except for the filling factors ${f}_{x}$ and ${f}_{z}$, which are the variables in this plot.Reuse & Permissions

###### Figure 6

The Casimir force between a gold half-space and an orthorhombic, slightly in-plane anisotropic MM for different values of the filling factors ${f}_{x}$ and ${f}_{y}$. The bands are characterized by a certain value of ${f}_{x}$, as shown in the legend, and a continuum of values of ${f}_{y}$, from ${f}_{y}=0.8{f}_{x}$ to ${f}_{y}=1.2{f}_{x}$. All the other parameters involved are exactly the same as those used in Fig. 4.Reuse & Permissions

###### Figure 7

The ratio $F\u2215{F}_{\mathrm{C}}$ between a gold half-space and an isotropic silver-based metamaterial for different values of the dissipation parameters. The main plot shows the effect of the simultaneous modification of electric and magnetic dissipation. Inset (a) shows the effect of electric dissipation alone for different values of the ratio ${\gamma}_{e}\u2215{\Omega}_{e}=0.1$ (solid), 0.5 (dashed), 2.5 (dotted). Inset (b)s shows the effect of magnetic dissipation alone for different values of the ratio ${\gamma}_{m}\u2215{\Omega}_{m}=0.1$ (solid), 0.5 (dashed), 2.5 (dotted). The filling factor is $f={10}^{-4}$ in all three plots, and all other parameters except the dissipation coefficients are the same as in Fig. 4.Reuse & Permissions

###### Figure 8

Temperature dependence of the Casimir force between a metallic plate and a metamaterial. We plot the Casimir force between a metamaterial and a Drude metal (a) or a plasma metal (b) for different temperatures. We stress that negative values of the force characterize repulsion, and that all parameters are the same as the ones used in the $f=0$ curve of Fig. 4.Reuse & Permissions

###### Figure 9

The ratio $F\u2215{F}_{\mathrm{C}}$ for a gold half-space facing an isotropic, nonconnected, and gold-based metamaterial. The parameters for the metal are ${\Omega}_{2,\mathrm{met}}\u2215\Omega =0.96$, ${\gamma}_{2,\mathrm{met}}\u2215\Omega =0.004$, and for the metamaterial we have ${\Omega}_{e}\u2215\Omega =0.34$, ${\Omega}_{m}\u2215\Omega =0.064$, ${\omega}_{e}\u2215\Omega =0.2$, ${\omega}_{m}\u2215\Omega =0.15$, ${\gamma}_{e}\u2215\Omega =0.04$, ${\gamma}_{m}\u2215\Omega =0.02$, $f=0.1$. The inset shows the permittivity and permeability inside the MM, as given by Eq. (67), but as functions of imaginary frequencies $\xi $.Reuse & Permissions

###### Figure 10

The permittivity ${\u03f5}_{\text{true}}\left(i\xi \right)$ and permeability ${\mu}_{\text{true}}\left(i\xi \right)$. The parameters are ${\u03f5}_{\infty}=2$, ${\Omega}_{\mathrm{pol}}\u2215\Omega =0.4$, ${\omega}_{\mathrm{pol}}\u2215\Omega =0.15$, ${\gamma}_{\mathrm{pol}}\u2215\Omega =0.001$.Reuse & Permissions

###### Figure 11

The plot of $\Delta P={P}^{\left(2\right)}-{P}^{\left(1\right)}$ for different temperatures and models. Following our conventions throughout the paper, a positive force means attraction. The parameters of ${P}^{\left(1\right)}$ are the same used in Fig. 8, in dimensional units they are ${\Omega}_{D}=1.32\times {10}^{16}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\gamma}_{D}=5.48\times {10}^{13}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$ (${\gamma}_{D}=0$ for the plasma curve), ${\Omega}_{e}=4.7\times {10}^{15}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\Omega}_{m}=8.7\times {10}^{14}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\omega}_{e}=2.7\times {10}^{15}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\omega}_{m}=2\times {10}^{15}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\gamma}_{e}=5.5\times {10}^{14}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\gamma}_{m}=2.7\times {10}^{14}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\Omega}_{2,1}=2.52\times {10}^{16}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\omega}_{2,1}=2.48\times {10}^{16}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\Omega}_{2,2}=6.4\times {10}^{15}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\omega}_{2,2}=3.8\times {10}^{15}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\Omega}_{2,3}={\omega}_{2,3}=1.9\times {10}^{14}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, ${\gamma}_{2,1}={\gamma}_{2,2}={\gamma}_{2,3}=0$, $f=0.1$, and the parameters of ${P}^{\left(2\right)}$ are exactly the same except for ${\Omega}_{e}=0$. The inset shows the same plot on a different scale, since in the larger one it is not possible to see $\Delta P$ for large distances.Reuse & Permissions

###### Figure 12

The difference in frequency shifts caused by the presence or absence of magnetic activity in the metamaterial. Everything is assumed to be at zero temperature. The MM is the same used in Fig. 11 and the parameters for the Rb atom are $m=1.45\times {10}^{-25}\phantom{\rule{0ex}{0ex}}\mathrm{Kg}$, ${\alpha}_{0}=4.74\times {10}^{-23}\phantom{\rule{0ex}{0ex}}{\mathrm{cm}}^{3}$, and ${\omega}_{0}=2.54\times {10}^{15}\phantom{\rule{0ex}{0ex}}\mathrm{rad}\u2215\mathrm{s}$, with an unperturbed trap frequency of ${\omega}_{z}=2\pi \times 229\phantom{\rule{0ex}{0ex}}\mathrm{Hz}$.Reuse & Permissions