Chiral Casimir forces: Repulsive, enhanced, tunable

Both theoretical interest and practical significance attach to the sign and strength of Casimir forces. A famous, discouraging no-go theorem states that “the Casimir force between two bodies with reflection symmetry is always attractive.” Here, we identify a promising way to avoid the assumptions of the no-go theorem, and propose a universal way to realize repulsive Casimir forces. We show that the sign and strength of Casimir forces can be adjusted by inserting optically active or gyrotropic media between bodies, and modulated by external fields.

Figure 1

Schematic illustration of chiral Casimir effect. Two parallel, uncharged plates ($A$ and $B$) are placed at a distance $l$ separated by chiral material $C$. The red dots and green dots represent right-circular polarized photons and left-circular polarized photons. The arrows indicate the propagating directions of chiral photons. $k_{R\left(L\right)}^{\pm }$ represent velocity of chiral photons, where superscript $\pm$ corresponds to their propagating directions, and the subscript $R/L$ correspond to their chirality.

Figure 2

Feynman diagrams for normal Casimir energy and chiral Casimir energy. (a) Shows the Feynman diagram representation for normal Casimir energy when chiral symmetry of photon is kept. Black wavy lines represent photon propagator ${\widehat{D}}_0$ and filled bubbles represent current-current correlation functions $T_A$ and $T_B$. (b) Shows the Feynman diagram representation for Casimir energy when chiral symmetry is broken, namely, the velocity of photons depend on their chirality. Red and green wavy lines correspond to Green's functions for right-circular polarized photons and left-circular polarized photons, respectively.

Figure 3

Chiral Casimir force due to Faraday effect, normalized to the original metallic Casimir force per area $F_0=-{\pi }^2\hslash c/\left(240l^4\right)$. (a) Shows the Casimir force enhancement in different magnetic field. The red and blue curves represent Casimir force at magnetic field $B=4\text{T}$ and $B=10\mathrm{T}$, respectively. The shadow region corresponds to repulsive Casimir force regime. (b) Shows how the magnetic field $B$ can control the Casimir force. The solid and dashed lines represent the Casimir force that is measured at the distance $l=8$ and $6\mu \mathrm{m}$, respectively.

Figure 4

Chiral Casimir force between two parallel plates mediated by an optically active material. Solid (dashed) curve corresponds to the ratio of Casimir force $F_c/F_0$ for frequency-dependent (-independent) specific rotation ${\alpha }_0$. Based on experimental results in optically active materials [20], the parameters are chosen as ${\omega }_p={10}^{16}{\mathrm{s}}^{-1}$, ${\gamma }_0=2\times {10}^6\mathrm{deg}{\mathrm{dm}}^{-1}{\mathrm{g}}^{-1}{\mathrm{cm}}^3$, ${\gamma }_1\approx {10}^{-21}\mathrm{deg}{\mathrm{dm}}^{-1}{\mathrm{g}}^{-1}{\mathrm{cm}}^3{\mathrm{s}}^2$, and $\rho =1\mathrm{g}{\mathrm{cm}}^{-3}$ in the calculation.

Figure 5

Temperature dependence of chiral Casimir force. Chiral Casimir force is calculated at three different temperatures $T=0$ K (black curve), $T=200$ K (blue curve), and $T=298$ K (red curve). Parameters are set as ${\omega }_p={10}^{16}{\mathrm{s}}^{-1}$, ${\gamma }_0=2\times {10}^6\mathrm{deg}{\mathrm{dm}}^{-1}{\mathrm{g}}^{-1}{\mathrm{cm}}^3$, ${\gamma }_1=0\mathrm{deg}{\mathrm{dm}}^{-1}{\mathrm{g}}^{-1}{\mathrm{cm}}^3{\mathrm{s}}^2$. The mass concentration is set as $\rho =1\mathrm{g}{\mathrm{cm}}^{-3}$ in (a) and $\rho =0\mathrm{g}{\mathrm{cm}}^{-3}$ in (b).

Figure 6

Schematic illustration of scattering process between two plates with a gyrotropic medium inserted. (a) Shows odd times of reflection, where the first reflection takes place at the left plate. For simplicity, we only show one time reflection case. (b) Shows even times of reflection, where the first reflection takes place at the right plate. Again, we only show the least two times of reflection case for simplicity.