Single-mode heat conduction by photons
  Electrical conductance is quantized in units of $\sigma_{\rm Q}=2e^2/h$ in ballistic one-dimensional conductors. Similarly, thermal conductance at temperature $T$ is expected to be limited by the quantum of thermal conductance of one mode, $G_{\rm Q} = \frac{\pi k_{\rm B}^2}{6\hbar}T$, when physical dimensions are small in comparison to characteristic wavelength of the carriers. The relation between $\sigma_{\rm Q}$ and $G_{\rm Q}$ obeys the Wiedemann-Franz law for ballistic electrons (apart from factor 2 in $\sigma_{\rm Q}$ due to spin degeneracy), but somewhat amazingly the same expression of $G_{\rm Q}$ is expected to hold also for phonons and photons, or any other particles with arbitrary exclusion statistics. The single-mode heat conductance is particularly relevant in nano-structures, e.g., when studying heat conduction by phonons in dielectric materials, or cooling of electrons in metals at very low temperatures. Here we show, based on our experimental results, that at low temperatures heat is transferred by photon radiation, in our case along a superconducting line, when electron-phonon as well as normal electronic heat conduction are frozen out. Thermal conductance is limited by $G_{\rm Q}$, approaching this value towards low temperatures. Our observation has implications on, e.g., performance and design of ultra-sensitive bolometers and electronic micro-refrigerators, whose operation is largely dependent on weak thermal coupling between the device and its environment. 
M. Meschke, W. Guichard, J.P. Pekola