Exceptional points treatments to nanoscale open quantum systems

We study the applicability of exceptional points at nanoscale. Non-Hermitian quantum mechanics is a popular way of treating open quantum systems which is employed in many fields of theoretical research from optics, opto-mechanics, and polaritonics, to quantum field theory, molecular physics, and quantum transport. The most non-trivial physics happens at and in vicinity of degeneracies of the spectrum of a non-Hermitian operator, exceptional points (EPs). We discuss exceptional points of a non-Hermitian Hamiltonian (HEPs) and Liouvilian (LEPs) using generic models (two vibrational modes in a cavity for HEPs and driven two-level system in a thermal environment for LEPs). Starting from the exact nonequilibrium Green's function (NEGF) description, we analyze approximations employed to arrive at either HEPs through formulation of an effective non-Hermitian Hamiltonian or LEPs through the Bloch quantum master equation (QME). In particular, we point to inconsistencies of the EP approaches in accounting for projections of self-energies (HEP includes retarded projection but disregards lesser/greater projections of the same self-energy; LEP includes lesser/greater projections but disregards retarded projection) and to the Markov character of the approaches. Limitations of the exceptional point treatments are illustrated with numerical simulations.