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Exceptional points (EPs) are degeneracies of classical and quantum opensystems, which are studied in many areas of physics including optics,optoelectronics, plasmonics, and condensed matter physics. In the semiclassicalregime, open systems can be described by phenomenological effectivenon-Hermitian Hamiltonians (NHHs) capturing the effects of gain and loss interms of imaginary fields. The EPs that characterize the spectra of suchHamiltonians (HEPs) describe the time evolution of a system without quantumjumps. It is well known that a full quantum treatment describing more genericdynamics must crucially take into account such quantum jumps. In a recent paper[F. Minganti $et$ $al.$, Phys. Rev. A $\mathbf{100}$, $062131$ ($2019$)], wegeneralized the notion of EPs to the spectra of Liouvillian superoperatorsgoverning open system dynamics described by Lindblad master equations.Intriguingly, we found that in situations where a classical-to-quantumcorrespondence exists, the two types of dynamics can yield different EPs. In arecent experimental work [M. Naghiloo $et$ $al.$, Nat. Phys. $\mathbf{15}$,$1232$ ($2019$)], it was shown that one can engineer a non-HermitianHamiltonian in the quantum limit by postselecting on certain quantum jumptrajectories. This raises an interesting question concerning the relationbetween Hamiltonian and Lindbladian EPs, and quantum trajectories. We discussthese connections by introducing a hybrid-Liouvillian superoperator, capable ofdescribing the passage from an NHH (when one postselects only thosetrajectories without quantum jumps) to a true Liouvillian including quantumjumps (without postselection). Beyond its fundamental interest, our approachallows to intuitively relate the effects of postselection and finite-efficiencydetectors.

Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories