Suppose we have two scalar fields $\varphi, \kappa$. Next, suppose there is a region in space where they are mix with each other, i.e., we have a lagrangian

$$

\tag 1

L_{\text{int}} = A \varphi \kappa

$$

By taking into account their kinetic term, we have following EOMS:

$$

\left(\omega^{2} + \partial_{\mathbf{r}}^2 - \begin{pmatrix}0 & A \\ A & 0\end{pmatrix}\right)\begin{pmatrix}\varphi\\ \kappa\end{pmatrix} = 0

$$

It gives rise to particle oscillations.

Next, suppose we have a beam of $\varphi$ particles propagating along $z$ axis. After entering the domain (say, at $z=0$) in which there is the interaction $(1)$ it begins to oscillate into $\kappa$ particle. I want to calculate the probability of oscillation at $z>0$. It turns out that it is proportional to

$$

P_{\varphi\to\kappa}\sim |e^{-ik_{+}z}-e^{-ik_{-}z}|, \quad k_{\pm} = \sqrt{\omega^2 \mp A}

$$

It turns out that for $|A|>\omega$ one of the momenta $k_{+}$, $k_{-}$ becomes imaginary, and the probability doesn't behave as oscillating function, but instead is exponentially amplified or damped.

What is the physical reason for this?