Circuit QED theory of direct and dual Shapiro steps with finite-size transmission line resonators

We investigate the occurrence of direct and dual Shapiro steps for a Josephson junction coupled to a finite-size transmission line resonator. We treat both problems through a circuit QED approach with a large, but finite number of photon modes. For the dual case, we do not assume the (approximate) charge-phase duality, but include the full multi-band dynamics for the Josephson junction. Mean-field equations within such Hamiltonian approach reproduce the result obtained through a dissipative classical equation when the number of transmission line modes is large enough. To account for quantum and thermal fluctuations, we go beyond the mean-field treatment within a truncated Wigner approach. The fluctuations are shown to modify both the direct and the dual steps. We show how the dual steps are very sensitive to these fluctuations and identify the key physical parameters for the junction and the transmission line controlling their robustness, which is essential for applications to close the quantum metrological triangle.