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Fractional Josephson effect versus fractional charge in superconducting-normal metal hybrid circuits

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.