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\begin{align*}
\ket{\Psi_{m}(x)}= &\cos\theta_m \left(   a_L \exp(-i \int_0^x \omega_m(x)/2 dx )  \ket{\nu_L(x)} + a_H \exp(i\int_0^x \omega_m(x)/2 dx) \ket{\nu_H(x)}   \right) \\
&+ \sin\theta_m \left( -a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx )  \ket{\nu_L(x)} + a_L^* \exp(i\int_0^x \omega_m(x)/2 dx) \ket{\nu_H(x)}  \right) \\
=& \left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x)/2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx )  \right)\ket{\nu_L(x)} \\
& +\left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m   a_L^* \exp(i\int_0^x \omega_m(x)/2 dx)  \right)\ket{\nu_H(x)} \\
=& \left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x) /2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx )  \right)( \cos\theta_m \ket{\nu_e} - \sin \theta_m \ket{\nu_x} ) \\
& +\left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m   a_L^* \exp(i\int_0^x \omega_m(x)/2 dx)  \right) ( \sin\theta_m \ket{\nu_e} + \cos\theta_m \ket{\nu_x} ) \\
=&  \left[\left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x)/2 dx ) - \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx )  \right) \cos\theta_m \right.\\
 &\left.  +  \left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) + \sin\theta_m   a_L^* \exp(i\int_0^x \omega_m(x)/2 dx)  \right) \sin\theta_m  \right]\ket{\nu_e} \\
 & + \left[ -\left(\cos\theta_m a_L\exp(-i \int_0^x \omega_m(x)/2 dx ) -  \sin\theta_m a_H^* \exp(-i \int_0^x \omega_m(x)/2 dx )  \right) \sin \theta_m  \right.\\
 & \left. + \left( \cos\theta_m a_H \exp(i\int_0^x \omega_m(x)/2 dx) +  \sin\theta_m   a_L^* \exp(i\int_0^x \omega_m(x)/2 dx)  \right) \cos\theta_m  \right] \ket{\nu_x} 
\end{align*}