Trace expressions and associated limits for nonequilibrium Casimir torque

We exploit fluctuational electrodynamics to present trace expressions for the torque experienced by arbitrary objects in a passive, nonabsorbing, rotationally invariant background environment. Specializing to a single object, this formalism, together with recently developed techniques for calculating bounds via Lagrange duality, is then used to derive limits on the maximum Casimir torque that a single object with an isotropic electric susceptibility can experience when out of equilibrium with its surrounding environment. The maximum torque achievable at any wavelength is shown to scale in proportion to body volumes in both subwavelength (quasistatics) and macroscopic (ray optics) settings, and come within an order of magnitude of achievable torques on topology optimized bodies. Finally, we discuss how to extend the formalism to multiple bodies, deriving expressions for the torque experienced by two subwavelength particles in proximity to one another.

Figure 1

Bounds on maximum angular momentum transfer based on the conservation of energy. An isolated body out of equilibrium can exchange net angular momentum with the environment, illustrated by an inset schematic. The figure shows the maximum spectral angular momentum transfer ${\mathrm{\Phi }}_J\left(\omega \right)$ (defined in the main text) at a single wavelength $\lambda$ allowed for a body enclosed in a spherical design volume of radius $R$, for multiple values of body material susceptibilities $\chi \left(\omega \right)$, captured by the material factor ${\parallel \chi \left(\omega \right)\parallel }^2/\text{Im}\left[\chi \left(\omega \right)\right].{\mathrm{\Phi }}_{J,max}\left(\omega \right)$ is seen to smoothly blend the intuitively expected $\propto V$ behaviors of quasistatic and ray-optic regimes, with an intermediate regime of growing radiative losses that suppresses the overall prefactor in the volumetric scaling. The full bounds, involving a relaxation of physics through the conservation of energy or optical theorem [30], described in Appendix pp6, are shown as dashed lines. Solid lines describe a semianalytical bound that, while looser owing to additional relaxations, provides an intuitive picture of the various wave contributions to torque (see main text). The dots indicate values of ${\mathrm{\Phi }}_J\left(\omega \right)$ observed in topology-optimized structures [31, 32, 33].

Figure 2

Torque on a particle due to fluctuating sources in a neighboring particle. Two objects can exchange net angular momentum among themselves and their environment, illustrated by an inset schematic. For concreteness, we plot ${\mathit{\tau }}_2^{\left(1\right)}\left(T_2\right)·{\mathbf{e}}_z$, the torque on particle 1 due to the sources in particle 2, normalized by $\hslash V_1{V}_2$ where $V_1$ and $V_2$ are the volumes of particle 1 and 2, respectively, as a function of the separation $d$ along the $x$ direction of two InSb particles in the $xy$ plane subject to an external magnetic field in the $z$ direction. Solid (dashed lines) indicate positive (negative) values.