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