  Fractionally charged excitations play a central role in condensed matter
physics, and can be probed in different ways. If transport occurs via
dissipation-less supercurrents, they manifest as a fractional Josephson effect,
whereas in dissipative transport they can be revealed by the transport
statistics. However, in a regime where supercurrents and lossy currents
coincide, a full understanding of the relationship between these two transport
phenomena is still missing. Moreover, especially for superconducting circuits,
the question of how noninteger quasicharges can be reconciled with charge
quantization is still not fully resolved, and plays an important role for the
circuit dynamics. Here, we aim to unify the above concepts by studying the
system-detector dynamics in terms of a Lindbladian capturing both coherent and
dissipative transport. Charge quantization is here a conserved property of the
detector basis of the Lindbladian, while charge fractionalization is a
topological property of its complex-valued eigenspectrum. We show that already
conventional superconductor-normal metal hybrid circuits exhibit a variety of
topological phases, including an open quantum system version of a fractional
Josephson effect. Surprisingly, quasiparticles, usually considered a
detrimental side effect, are here a necessary ingredient to observe nontrivial
transport behaviour.
