Violation of detailed balance in microwave circuits: theory and experiment

We propose an approach for addressing detailed balance violation in electrical circuits based on the scattering matrix formalism commonly used in microwave electronics. This allows the inclusion of very simple retardation effects, which are paramount at high frequencies. We define the spectral densities of phase space angular momentum, heat transfer and cross power, which can serve as criteria for detailed balance violation. We confirm our theory with measurements in the 4-8 GHz frequency range on several two-port circuits of varying symmetries, in space and time. The restults validate our approach and enable the straightforward treatment of quantum circuits at ultra-low temperatures.