Metrics for the violation of detailed balance in microwave circuits: theory and experiment

We propose a new approach to detailed balance violation in electrical circuits by relying on the scattering matrix formalism commonly used in microwave electronics. This allows to 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 will allow to treat quantum circuits at ultra-low temperature