\documentclass{article}
\usepackage{amsmath}


\begin{document}


\begin{eqnarray}
I = \sum_{\rm{ext \: spins}} \left| \left< out \left| \hat{T} \right| in \right> \right|^{2} 
= \sum_{\rm{ext \: spins}} \left< out \left| \hat{T} \right| in \right> 
\left< in \left| \hat{T}^* \right| out \right> \nonumber
\end{eqnarray}

Define the spin density matrix $\rho$ such that:
\begin{eqnarray}
\left| in \left> \right< in \right| =  
\sum_{i,j} \rho_{i,j} = \sum_{i,j} \left| i \left> \right< j \right|  \nonumber
\end{eqnarray}

Now

\begin{eqnarray}
I = \sum_{\rm{ext \: spins}}\sum_{i,j} \left< out \left| ^i\hat{T} \rho_{i,j} 
 {^j\hat{T}^*} \right| out \right> \nonumber
\end{eqnarray}

Let $\hat{T} = \hat{T_d}\hat{T_p}$, where subscript $d \equiv$ decay and $p \equiv$ production 

\begin{eqnarray}
I = \sum_{\rm{ext \: spins}}\sum_{i,j} \left< out \left| \hat{T}_d ^i\hat{T}_p \rho_{i,j}
{^j\hat{T}^*_p} \hat{T}^*_d \right| out \right> \nonumber
\end{eqnarray}

Insert

\begin{eqnarray}
1 = \sum_{X}\left| X \left>  \right< X \right| \nonumber
\end{eqnarray}

and

\begin{eqnarray}
1 = \sum_{X^{\prime}}\left|X^{\prime} \left> \right< X^{\prime} \right| \nonumber
\end{eqnarray}


\begin{eqnarray}
I = \sum_{\rm{ext \: spins}}\sum_{i,j}\sum_{X,X^{\prime}} 
\left<out\right|\hat{T}_d\left|X\right>\left<X\right|^i\hat{T}_p\rho_{i,j}{^j\hat{T}^*_p}\left|X^{\prime} 
\right> \left< X^{\prime} \right|\hat{T}^*_d \left| out \right> \nonumber
\end{eqnarray}


\begin{eqnarray}
I(\Omega,\Phi) = 2 \kappa \sum_k \left\{ 
(1 - P_{\gamma} ) \left|  \sum_{l,m} [l]^{(-)}_{m;k} Re[Z^m_l(\Omega,\Phi)]  \right|^2 +
(1 - P_{\gamma} ) \left|  \sum_{l,m} [l]^{(+)}_{m;k} Im[Z^m_l(\Omega,\Phi)]  \right|^2 + \right. \nonumber \\
\left.
(1 + P_{\gamma} ) \left|  \sum_{l,m} [l]^{(+)}_{m;k} Re[Z^m_l(\Omega,\Phi)]  \right|^2 +
(1 + P_{\gamma} ) \left|  \sum_{l,m} [l]^{(-)}_{m;k} Im[Z^m_l(\Omega,\Phi)]  \right|^2 
 \right\}
\end{eqnarray}

If $P = 0$

\begin{eqnarray}
I(\Omega,\Phi) = 2 \kappa \sum_k \left\{ 
\left|  \sum_{l,m} [l]^{(-)}_{m;k} Z^m_l(\Omega,\Phi)  \right|^2 +
\left|  \sum_{l,m} [l]^{(+)}_{m;k} Z^m_l(\Omega,\Phi)  \right|^2
 \right\}
\end{eqnarray}

Let $r = (-), (+)$ then


\begin{eqnarray}
I(\Omega,\Phi) = 2 \kappa \sum_{k,r} \left\{ 
\left|  \sum_{l,m} [l]^{r}_{m;k} Z^m_l(\Omega,\Phi)  \right|^2
 \right\}
\end{eqnarray}


\end{document}
