Instead, we shall focus on a fundamental difference between both theories, which is the unitary versus pseudo-unitary time evolution of the respective fermionic and bosonic fields. Its consequences on the resulting Fraunhofer-like enhancements described above will be studied in this paper in great detail.

Note that our work fits nicely in the upgrowing area of research devoted to non-Hermitian quantum theories. The reason being that in the case of scalar pair production, the dynamics of a single eigenmode of the bosonic field is determined by a pseudo-Hermitian Hamiltonian \[Eq. \]. Thus, our work adds to a long list of potential applications of pseudo-Hermitian theories that includes generalized coherent states [@Wodkiewicz], synthetic optical lattices [@Makris; @Guo], waveguide couplers [@Ruter; @Lee], laser cavities [@Peng; @Feng], or Rabi systems [@Torosov; @Kus] (for more application, see, also [@Rotter1; @Moiseyev; @Rotter2]). Interestingly, in this context the role of the so-called [*exceptional points*]{} is frequently studied [@Peng; @Feng; @Praxmeyer]. While non-Hermitian Hamiltonians have complex eigenvalues, at those points their eigenvalues coalesce. In other words, they exhibit a [*non-avoided crossing*]{} where their real components are identical, as are their imaginary ones. This leads to counterintuitive effects when steering the system in the vicinity of the exceptional points 