Phase-charge duality in Josephson junction circuits: Role of inertia and effect of microwave irradiation

We investigate the physics of coherent quantum phase slips in two distinct containing small Josephson junctions: (i) a single junction embedded an inductive environment and (ii) a long chain of junctions. Starting from standard Josephson Hamiltonian, the single junction circuit can be analyzed quasi-classical methods; we formulate the conditions under which the quasi-charge dynamics is exactly dual to the usual phase dynamics with Josephson tunneling. For the chain we use the fact that its behavior can be characterized by one variable: the number m of phase slips present on it. We conclude that the dynamics of the quasi-charge is again exactly dual to the standard phase dynamics of single Josephson junction. In both cases we elucidate the role of the, essential to obtain exact duality. These conclusions have profound for the behavior of single junctions and chains under microwave. Since both systems are governed by a model exactly dual to the resistively and capacitively shunted junction model, we expect the of current-Shapiro steps. We numerically calculate the corresponding-voltage characteristics in a wide range of parameters. Our results are interest in view of a metrological current standard.