Nonlocal microscopic theory of Casimir forces at finite temperature

The interaction energy between two metallic slabs in the retarded limit at finite temperature is expressed in terms of surface polariton propagators for separate slabs, avoiding the usual matching procedure, with both diamagnetic and paramagnetic excitations included correctly. This enables appropriate treatment of arbitrary electron density profiles and fully nonlocal electronic response, including both collective and single-particle excitations. The results are verified by performing the nonretarded and long-wavelength (local) limits and showing that they reduce to the previously obtained expressions. Possibilities for practical use of the theory are explored by applying it to calculation of various contributions to the Casimir energy between two silver slabs.

Figure 1

Feynman’s diagram for $S_k^{\text{dif con}}$, i.e., the $n$th term in the perturbation expansion.

Figure 2

Interobject fluctuation diagrams in terms of current-current response functions.

Figure 3

(a) Expansion of the current-current response function in terms of $D^S$; (b) irreducible polarizabilities; (c) two-photon exchange diagram.

Figure 4

Geometry of the system.

Figure 5

Propagators of the electromagnetic fields.

Figure 6

Feynman diagrams that represent induced electrical fields on (a) left and (b) right inner metal surfaces.

Figure 7

Thin solid line: rectangular electronic density profile that corresponds to jellium edges. Thick solid line: LDA electron density profile. The corresponding parameters are presented in the text.

Figure 8

Casimir free energies for two silver slabs with thicknesses $31a_0$ at room temperature ($T=300$ K) calculated in the jellium model, as a function of distance between the slabs. Solid (black) lines: Drude model with dissipation; dashed (red) lines: nonlocal calculation without dissipation; dotted (blue) lines: Drude model without dissipation. Graph (a) shows distances up to $1\hspace{0.5em}\mu$m while graph (b) shows distances from 1 to $5\hspace{0.5em}\mu$m. The four graphs in the bottom show $s(\text{TE})$ and $p(\text{TM})$ contributions of the Casimir energies of the graphs above them.