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