Trace formulas for nonequilibrium Casimir interactions, heat radiation, and heat transfer for arbitrary objects

We present a detailed derivation of heat radiation, heat transfer, and (Casimir) interactions for $N$ arbitrary objects in the framework of fluctuational electrodynamics in thermal nonequilibrium. The results can be expressed as basis-independent trace formulas in terms of the scattering operators of the individual objects. We prove that heat radiation of a single object is positive, and that heat transfer (for two arbitrary passive objects) is from the hotter to a colder body. The heat transferred is also symmetric, exactly reversed if the two temperatures are exchanged. Introducing partial wave expansions, we transform the results for radiation, transfer, and forces into traces of matrices that can be evaluated in any basis, analogous to the equilibrium Casimir force. The method is illustrated by (re)deriving the heat radiation of a plate, a sphere, and a cylinder. We analyze the radiation of a sphere for different materials, emphasizing that a simplification often employed for metallic nanospheres is typically invalid. We derive asymptotic formulas for heat transfer and nonequilibrium interactions for the cases of a sphere in front a plate and for two spheres, extending previous results. As an example, we show that a hot nanosphere can levitate above a plate with the repulsive nonequilibrium force overcoming gravity, an effect that is not due to radiation pressure.

Figure 1

The results derived in the previous sections (Secs. 3, 4, 5, 6) are completely general and apply, e.g., to the configuration on the left hand side. The right hand side shows a configuration that allows representation in partial waves as derived in this section, because the enclosing spheres or ellipsoids do not overlap.

Figure 2

Energy emitted by SiO${}_2$ (upper curves) and gold (lower curves) spheres at $T=300$ K in a cold environment as a function of radius $R$, normalized by the Stefan-Boltzmann result [Eq. (126) with $\epsilon \left(T\right)=1$]. Solid lines correspond to the exact result from Eq. (124) (for SiO${}_2$, the data are from Ref. 28); different approximations valid for small spheres are depicted by long dashes correspond to Eq. (129), short dashes to Eq. (131) omitting the term $|{\mathcal{T}}_1^P{|}^2$, and dots to Eq. (131).

Figure 3

The system of two spheres in thermal nonequilibrium.

Figure 4

The system of a sphere in front a plate in thermal nonequilibrium.

Figure 5

The net force (including gravity) on a spherical aluminum shell (outer and inner radii of 73 and 23 nm) below a SiC plate, for $T_p=300$ K and $T_{\mathrm{env}}=2862$ K (${\lambda }_{T_{\mathrm{env}}}=800$ nm). ($T_s$ plays no visible role.) The force is normalized by the gravitational force $F_G$, with positive values pointing up towards the plate (i.e., gravity gives a negative contribution). The point of zero force at smaller $d$ is stable; the inset shows oscillations around this point when the sphere is initially put at $d=-4$ $\mu$m with zero velocity.

Figure 6

Inset: The net force on a dielectric sphere of radius 60 nm above a dielectric plate, including the gravitational force $F_G$ on the sphere (which is negative) for $T_p=T_{\mathrm{env}}=300$ K and $T_s=916$ K. The strong nonequilibrium repulsion is due to near-field effects, not radiation pressure, and leads to a stable zero force point (including gravity). The dashed line shows the total force for $T_p=T_{\mathrm{env}}=T_s=300$ K. The main figure shows the trajectory of the sphere, when dropped from 800 nm. Points show the trajectory including cooling down of the sphere, where the length of the vertical bars is proportional to $T_s\left(t\right)-300$ K in arbitrary units.