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Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the Scully-Lamb laser theory

In the past few decades, many works have been devoted to the study of exceptional points (EPs), i.e., exotic degeneracies of non-Hermitian systems. The usual approach in those studies involves the introduction of a phenomenological effective non-Hermitian Hamiltonian (NHH), where the gain and losses are incorporated as the imaginary frequencies of fields and from which the Hamiltonian EPs (HEPs) are derived. Although this approach can provide valid equations of motion for the fields in the classical limit, its application in the derivation of EPs in the quantum regime is questionable. Recently, a framework [Minganti {\it et al.}, \href{https://doi.org/10.1103/PhysRevA.100.062131}{Phys. Rev. A {\bf 100}, 062131 (2019)}], which allows one to determine quantum EPs from a Liouvillian EP (LEP), rather than from an NHH, has been proposed. Compared to the NHHs, a Liouvillian naturally includes quantum noise effects via quantum-jump terms, thus allowing one to consistently determine its EPs purely in the quantum regime. In this work we study a non-Hermitian system consisting of coupled cavities with unbalanced gain and losses, where the gain is far from saturation, i.e, the system is assumed to be linear. We apply both formalisms, based on an NHH and a Liouvillian within the Scully-Lamb laser theory, to determine and compare the corresponding HEPs and LEPs in the semiclassical and quantum regimes. {Our results indicate that, although the overall spectral properties of the NHH and the corresponding Liouvillian for a given system can differ substantially, their LEPs and HEPs occur for the same combination of system parameters.}