Oscillation probabilities for a PT-symmetric non-Hermitian two-state system

Abstract

There is growing interest in viable quantum theories with PT-symmetric non-Hermitian Hamiltonians, but a formulation of transition matrix elements consistent with positivity and perturbative unitarity has so far proved elusive. This Letter provides such a formulation, which relies crucially on the ability to span the state space in such a way that the interaction and energy eigenstates are orthonormal with respect to the same positive-definite inner product. We apply this non-Hermitian approach to two-neutrino flavour oscillations, and show how it can accommodate the seesaw mechanism.

Figures (2)

• Figure 1: Comparison of the transition and survival probabilities (a) and squared eigenmasses (b) for the Hermitian and non-Hermitian models as a function of the parameter $\eta$. For the Hermitian case, the mass eigenvalues diverge for large $\eta$, with the lower eigenvalue crossing zero and becoming negative at a value of $\eta^2=(m_1^2+m_2^2)^2/(m_1^2-m_2^2)^2-1$. This occurs before the transition and survival probabilities saturate. For the non-Hermitian case, the mass eigenvalues merge at the exceptional point $\eta=1$, at which the probabilities saturate.
• Figure 2: Polar plot of the Dirac norm $r(\vartheta)/r(\pi)$ for different values of the parameter $\eta$: that for $\eta=0.1$ (dashed, blue) is close to a unit circle, centered on the origin; that for $\eta=0.5$ (dot-dashed, yellow) clearly deviates from the unit circle; and that for $\eta=0.9$ (solid, green) has a distinctive cardioid shape.