Chern number of Andreeev bound states with quasi-particle poisoning

Topological classification of closed quantum systems, known as the ten-fold way, is a well established formalism by now. On the other hand the generalization of this classification to open quantum systems is still a matter of ongoing research, with many different approaches. Recently, a classification for fermionic systems governed by quadratic Lindbladians into ten non-Hermitian Bernard-LeClair (BL) symmetry classes was proposed, which reduce to Altland-Zirnbauer classes in the closed limit. Here, we test and apply this classification scheme for multi-terminal Josephson junctions, which have the special property of harbouring a closed system Chern number in spite of all the materials being trivial. For this circuit, it is known that quasiparticle poisoning destroys the closed system topology, as it drives the system out of the topological subspace. In this work we show that a classification of the circuit in presence of quasiparticle poisoning is possible along the lines of the BL symmetry classes. We identify a generalized Chern number, which retains its quantization in spite of the poisoning. Investigations into how this Chern number shows up in the observables are currently underway.