  Transport measurements are fundamental for understanding condensed matter
phenomena, from superconductivity to the fractional quantum Hall effect.
Analogously, they can be powerful tools for probing synthetic quantum matter in
quantum simulators. Here we demonstrate the measurement of in-situ particle
current in a superconducting circuit lattice and apply it to study transport in
both coherent and bath-coupled lattices. Our method utilizes controlled
tunneling in a double-well potential to map current to on-site density,
revealing site-resolved current and current statistics. We prepare a strongly
interacting Bose-Hubbard lattice at different lattice fillings, and observe the
change in current statistics as the many-body states transition from superfluid
to Mott insulator. Furthermore, we explore non-equilibrium current dynamics by
coupling the lattice to engineered driven-dissipative baths that serve as
tunable particle source and drain. We observe steady-state current in discrete
conduction channels and interaction-assisted transport. These results establish
a versatile platform to investigate microscopic quantum transport in
superconducting circuits.
