  Superconducting circuits with nonlinear elements such as Josephson tunnel
junctions or kinetic inductance nanowires are the workhorse for microwave
quantum and superconducting sensing technologies. For devices, which can be
operated at high temperatures and large magnetic fields, nano-constrictions as
nonlinear elements are recently under intense investigation. Constrictions,
however, are far less understood than conventional Josephson tunnel junctions,
and their current-phase-relationships (CPRs) -- although highly important for
device design -- are hard to predict. Here, we present a niobium microwave
cavity with a monolithically integrated, neon-ion-beam patterned
three-dimensional (3D) nano-constriction. By design, we obtain a
DC-current-tunable microwave circuit and characterize how the
bias-current-dependent constriction properties impact the cavity resonance.
Based on the results of these experiments, we reconstruct the CPR of the
nanoconstriction. Finally, we discuss the Kerr nonlinearity of the device, a
parameter important for many high-dynamic-range applications and an
experimental probe for the second and third derivatives of the CPR. Our
platform provides a useful method to comprehensively characterize nonlinear
elements integrated in microwave circuits and could be of interest for current
sensors, hybrid quantum systems and parametric amplifiers. Our findings
furthermore contribute to a better understanding of nano-fabricated 3D
constrictions.
