Calculation of nonzero-temperature Casimir forces in the time domain

We show how to compute Casimir forces at nonzero temperatures with time-domain electromagnetic simulations, for example, using a finite-difference time-domain (FDTD) method. Compared to our previous zero-temperature time-domain method, only a small modification is required, but we explain that some care is required to properly capture the zero-frequency contribution. We validate the method against analytical and numerical frequency-domain calculations, and show a surprising high-temperature disappearance of a nonmonotonic behavior previously demonstrated in a pistonlike geometry.

Figure 1

Comparison between FDTD (red circles) and the analytical Lifshitz formula [25] (blue line) for the Casimir force between perfect-metal plates in one dimension with separation $a$. The $\omega =0$ and $\omega \ne 0$ contributions to the Matsubara sum (1) are plotted separately, in addition to the total force. The straightforward method of including the $\coth (\hslash \omega /2\mathit{kT})$ Bose-Einstein factor in the FDTD integration (green dashed line) gives an incorrect result because the $\omega =0$ pole requires special handling.

Figure 2

Comparison between FDTD (circles and diamonds) and BEM frequency domain (solid and dashed lines) calculation of the 2D Casimir force ($z$-invariant fluctuations) between two perfect-metal sidewalls (separation $d$), normalized by the proximity force approximation for the 2D parallel plates $F_{\mathrm{PFA}}=\hslash c\zeta (3)/8\pi a^2$. At $T=0$ (circles and solid lines) total force (black) varies nonmonotonically with $d$, due to competition between TE (red) and TM (blue) polarizations [21]. At $T=1\times \pi c\hslash /k_{\mathrm{B}}a$ (dashed lines and diamonds) BEM and FDTD match, but the nonmonotonicity disappears.