  We present a generic theoretical framework to describe non-reciprocal
microwave circulation in a multimode cavity magnonic system and assess the
optimal performance of practical circulator devices. We show that high
isolation (> 56 dB), extremely low insertion loss (< 0.05 dB), and flexible
bandwidth control can be potentially realized in high-quality-factor
superconducting cavity based magnonic platforms. These circulation
characteristics are analyzed with materials of different spin densities. For
high-spin-density materials such as yttrium iron garnet, strong coupling
operation regime can be harnessed to obtain a broader circulation bandwidth. We
also provide practical design principles for a highly integratible
low-spin-density material (vanadium tetracyanoethylene) for narrow-band
circulator operation, which could benefit noise-sensitive quantum microwave
measurements. This theory can be extended to other coupled systems and provide
design guidelines for achieving tunable microwave non-reciprocity for both
classical and quantum applications.
