  Microwave circulators play an important role in quantum technology based on
superconducting circuits. The conventional circulator design, which employs
ferrite materials, is bulky and involves strong magnetic fields, rendering it
unsuitable for integration on superconducting chips. One promising design for
an on-chip superconducting circulator is based on a passive Josephson-junction
ring. In this paper, we consider two operational issues for such a device:
circuit tuning and the effects of quasiparticle tunneling. We compute the
scattering matrix using adiabatic elimination and derive the parameter
constraints to achieve optimal circulation. We then numerically optimize the
circulator performance over the full set of external control parameters,
including gate voltages and flux bias, to demonstrate that this
multi-dimensional optimization converges quickly to find optimal working
points. We also consider the possibility of quasiparticle tunneling in the
circulator ring and how it affects signal circulation. Our results form the
basis for practical operation of a passive on-chip superconducting circulator
made from a ring of Josephson junctions.
