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Exceptional points (EPs) correspond to degeneracies of open systems. Theseare attracting much interest in optics, optoelectronics, plasmonics, andcondensed matter physics. In the classical and semiclassical approaches,Hamiltonian EPs (HEPs) are usually defined as degeneracies of non-HermitianHamiltonians such that at least two eigenfrequencies are identical and thecorresponding eigenstates coalesce. HEPs result from continuous, mostly slow,nonunitary evolution without quantum jumps. Clearly, quantum jumps should beincluded in a fully quantum approach to make it equivalent to, e.g., theLindblad master-equation approach. Thus, we suggest to define EPs viadegeneracies of a Liouvillian superoperator (including the full Lindbladianterm, LEPs), and we clarify the relations between HEPs and LEPs. We prove twomain theorems: Theorem 1 proves that, in the quantum limit, LEPs and HEPs musthave essentially different properties. Theorem 2 dictates a condition underwhich, in the ``semiclassical'' limit, LEPs and HEPs recover the sameproperties. In particular, we show the validity of Theorem 1 studying systemswhich have (1) an LEP but no HEPs, and (2) both LEPs and HEPs but for shiftedparameters. As for Theorem 2, (3) we show that these two types of EPs becomeessentially equivalent in the semiclassical limit. We introduce a series ofmathematical techniques to unveil analogies and differences between the HEPsand LEPs. We analytically compare LEPs and HEPs for some quantum andsemiclassical prototype models with loss and gain.

Quantum exceptional points of non-Hermitian Hamiltonians and Liouvillians: The effects of quantum jumps