Analogue Hawking radiation as a tunneling in a two-level $\mathcal{PT}$-symmetric system

In the light of a general scenario of a two-level non-Hermitian $\mathcal{PT}$-symmetric Hamiltonian we apply the tetrad-based method to analyze the possibility of analogue Hawking radiation. It is done by making use of the conventional null-geodesic approach wherein the associated Hawking radiation is described as a quantum tunneling process across a classically forbidden barrier which the event horizon imposes. An interesting aspect of our result is that our estimate for the tunneling probability is independent of the non-Hermitian parameter that defines the guiding Hamiltonian.