Two flavor neutrino oscillations in presence of non-Hermitian dynamics

We develop a consistent mathematical framework for studying two flavor neutrino oscillations in presence of non-Hermitian dynamics. We consider two approaches : (a) bi-orthonormal inner product defined by a positive-definite metric operator $\mathcal{G}$ and (b) the density matrix prescription by Brody and Graefe [Phys. Rev. Lett. 109, 230405 (2012)]. For the $\mathcal{P}\mathcal{T}$-symmetric case, we show that the $\mathcal{G}$ metric approach does not work well (probabilities are not conserved) both in $\mathcal{P}\mathcal{T}$-unbroken as well as $\mathcal{P}\mathcal{T}$-broken regime. Hence, we adopt the density matrix prescription by Brody and Graefe which is a positive semi-definite map. In the density matrix prescription, we note that probability in the steady state limit is not necessarily $1\mathrm{/}2$ thereby indicating non-Markovian behavior.