  We consider the dynamics of a quantum phase-slip junction (QPSJ) -- a dual
Josephson junction -- connected to a microwave source with frequency
$\omega_\textrm{mw}$. With respect to an ordinary Josephson junction, a QPSJ
can sustain dual Shapiro steps, consisting of well-defined current plateaus at
multiple integers of $ e \omega_\textrm{mw} / \pi$ in the current-voltage (I-V)
characteristic. The experimental observation of these plateaus has been elusive
up to now. We argue that thermal as well as quantum fluctuations can smear the
I-V characteristic considerably. In order to understand these effects, we study
a current-biased QPSJ under microwave irradiation and connected to an inductive
and resistive environment. We find that the effect of these fluctuations are
governed by the resistance of the environment and by the ratio of the
phase-slip energy and the inductive energy. Our results are of interest for
experiments aimed at the observation of dual Shapiro steps in QPSJ devices for
the definition of a new quantum current standard.

Mesoscale and Nanoscale Physics