Circuit quantum electrodynamics 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 there is (approximate) charge-phase duality, but include the full multiband dynamics for the Josephson junction. Mean-field equations within such a 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 steps 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.