Beyond-proximity-force-approximation Casimir force between two spheres at finite temperature. II. Plasma versus Drude modeling, grounded versus isolated spheres

A recent experiment [J. L. Garrett, D. A. T. Somers, and J. N. Munday, Phys. Rev. Lett 120, 040401 (2018)] measured for the first time the gradient of the Casimir force between two gold spheres in vacuum at room temperature, and placed a bound on the magnitude of the deviation of the measured force from the proximity force approximation (PFA). The present work extends a previous theoretical analysis of this experiment [G. Bimonte, Phys. Rev. D 97, 085011 (2018)], by analyzing in detail how the magnitude of the deviation from PFA is affected by the inclusion or neglect of ohmic dissipation at zero frequency, a much debated issue in the current Casimir literature, which goes by the name of the Drude vs plasma controversy. We analyze as well the effect of connecting the conductors to charge reservoirs, which is the standard configuration used in Casimir experiments. We describe a simple and effective decimation procedure, allowing for a faster computation of the Casimir force for large aspect ratios of the system.

Figure 1

The sphere-sphere Casimir setup. The sphere-sphere geometry is characterized by the effective radius $\tilde{R}=R_1{R}_2/(R_1+R_2)$ and the dimensionless parameter $u={\tilde{R}}^2/R_1{R}_2$.

Figure 2

The two-sphere system in bispherical coordinates. The thin solid and dashed lines are curves of constant bispherical coordinates $\mu$ and $\eta$ respectively. The sphere of radius $R_1$ has equation $\mu ={\mu }_{+}$, with ${\mu }_{+}>0$, while the sphere of radius $R_2$ has equation $\mu ={\mu }_{-}$, with ${\mu }_{-}<0$. Bispherical partial waves have their origins at the foci $F_{\pm }$, while standard spherical multipoles have their origins at the sphere centers $C_{\pm }$.

Figure 3

The upper panel shows the ratio between the zero-frequency contribution to the Casimir force and the total Casimir force, while the lower planel displays the analogous ratio for the force gradient. All forces are computed for gold at room temperature, using the PFA. Solid lines are for the Drude model, dashed lines for the plasma model.

Figure 4

Coefficients ${\beta }_{n>0}$ (upper panel) and ${\tilde{\beta }}_{n>0}$ (lower panel), computed using the DE. Solid and dashed lines correspond, respectively, to the Drude and plasma prescriptions. Red curves are for the sphere-plate geometry, blue curves for two spheres of equal radii.

Figure 5

Coefficients ${\beta }_{n=0}$ (upper panel) and ${\tilde{\beta }}_{n=0}$ (lower panel) for the Drude model, with grounded conductors. Solid lines are for two identical spheres with radius $R=40\mu \mathrm{m}$, dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=80\mu \mathrm{m}$, and the dot-dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=200\mu \mathrm{m}$. The thin horizontal blue lines represents the prediction of the DE.

Figure 6

Coefficients ${\beta }_{n=0}$ (upper panel) and ${\tilde{\beta }}_{n=0}$ (lower panel) for the Drude model, with isolated conductors. Solid lines are for two identical spheres with radius $R=40\mu \mathrm{m}$, dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=80\mu \mathrm{m}$, and the dot-dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=200\mu \mathrm{m}$.

Figure 7

Coefficients ${\beta }_{n=0}$ (upper panel) and ${\tilde{\beta }}_{n=0}$ (lower panel) for perfect grounded conductors. Solid lines are for two identical spheres with radius $R=40\mu \mathrm{m}$, dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=80\mu \mathrm{m}$, and the dot-dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=200\mu \mathrm{m}$.

Figure 8

Coefficients ${\beta }_{n=0}$ (upper panel) and ${\tilde{\beta }}_{n=0}$ (lower panel) for a perfect isolated conductor. Solid lines are for two identical spheres with radius $R=40\mu \mathrm{m}$, dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=80\mu \mathrm{m}$, and the dot-dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=200\mu \mathrm{m}$.

Figure 9

Coefficients $\beta$ (upper panel) and $\tilde{\beta }$ (lower panel) for grounded conductors. Solid lines are for two identical spheres with radius $R=40\mu \mathrm{m}$, dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=80\mu \mathrm{m}$, and the dot-dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=200\mu \mathrm{m}$.

Figure 10

Coefficients $\beta$ (upper panel) and $\tilde{\beta }$ (lower panel), for isolated conductors. Solid lines are for two identical spheres with radius $R=40\mu \mathrm{m}$, dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=80\mu \mathrm{m}$, and the dot-dashed lines are for two spheres of radii $R_1=40\mu \mathrm{m}$ and $R_2=200\mu \mathrm{m}$.

Figure 11

Room temperature Casimir force between two identical gold sphere of radius $R=40\mu \mathrm{m}$, normalized by the PFA force $F_{\mathrm{PFA}}^{(\mathrm{id})}$ for ideal spheres at zero temperature (upper panel), and the corresponding force gradient normalized by the force gradient ${F^{\prime }}_{\mathrm{PFA}}^{(\mathrm{id})}$ for ideal spheres at zero temperature (lower panel). Solid lines are for grounded spheres, dashed lines are for isolated spheres, while dot-dashed lines represent the standard PFA.