  We present a detailed report of microwave irradiation of ultra-narrow
superconducting nanowires. In our nanofabricated circuits containing a
superconducting NbSi nanowire, a dc blockade of current flow was observed at
low temperatures below a critical voltage Vc, a strong indicator of the
existence of quantum phase-slip (QPS) in the nanowire. We describe the results
of applying microwaves to these samples, using a range of frequencies and both
continuous-wave and pulsed drive, in order to search for dual Shapiro steps
which would constitute an unambiguous demonstration of quantum phase-slip. We
observed no steps, and our subsequent thermal analysis suggests that the
electron temperature in the series CrO resistors was significantly elevated
above the substrate temperature, resulting in sufficient Johnson noise to wash
out the steps. To understand the system and inform future work, we have
constructed a numerical model of the dynamics of the circuit for dc and ac bias
(both continuous wave and pulsed drive signals) in the presence of Johnson
noise. Using this model, we outline important design considerations for device
and measurement parameters which should be used in any future experiment to
enable the observation of dual Shapiro steps at experimentally accessible
temperatures and thus lead to the development of a QPS-based quantum current
standard.
