Violation of detailed balance in microwave circuits: theory and experiment

We propose an approach to detailed balance violation in electrical circuits based on the scattering matrix formalism commonly used in microwave electronics. This allows us to easily include 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. This validates our approach, which enables straightforward treatment of quantum circuits at ultra-low temperature.