Role of the quasi-particles in an electric circuit with Josephson junctions

Josephson junctions provide highly non-linear impedances at the root of many applications such as quantum limited parametric amplifiers or superconducting qubits. These junctions are often described by a sinusoidal relation I=Icsinφ which relates the current I to the integral over time of the voltage across the junction φ. This relation properly captures the contribution of the superconducting condensate but not the quasi-particles that appear when the system is driven out-of-equilibrium. Here, we construct a unifying framework that includes a microscopic description of the junction (full fledged treatment of the time-dependent Bogoliubov-De-Gennes equation) in presence of a classical electronic circuit. Our approach generalizes the standard Resistor-Capacitor-Josephson model (RCJ) to arbitrary junctions (including e.g. multi-terminal geometries and/or junctions that embed topological or magnetic elements) and classical circuits. We apply our technique to two situations. First, a RC circuit connected to single channel Josephson junction that exhibits Multiple Andreev Reflection (MAR) phenomena. We show that the theory properly describes both MAR and the hysteresis loops due to the electromagnetic environment. We show that out-of-equilibrium, the current-phase relation of the junction becomes strongly distorted from the simple sinusoidal form. Second, we embed the junction into a RLC circuit and show that the out-of-equilibrium non-sinusoidal current phase relation leads to a strong change of the shape of the resonance.