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\begin{equation}p_j(t)=\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{1,k}e^{ij\tilde{k}}\right|^2+\frac{1}{N^2}\left|\sum_{k=0}^{N-1}c_{2,k}(t)e^{ij\tilde{k}}\right|^2\end{equation}

where,

\begin{equation}c_{1,k}(t)=\cos\theta_kt-\frac{i\cos\tilde{k}\sin\theta_kt}{\sqrt{1+cos^2\tilde{k}}},

\end{equation}

\begin{equation}

c_{2,k}(t)=-\frac{ie^{i\tilde{k}}\sin\theta_kt}{\sqrt{1+cos^2\tilde{k}}}.

\end{equation}

\begin{equation}

\tilde{k}=\frac{2\pi k}{N}.

\end{equation}

\begin{equation}\sin\theta_k=\frac{1}{\sqrt{2}}\sin\frac{2\pi k}{N}.\end{equation}

Can anyone please help me find the laplace transform of $p_j(t)$? I would also be glad if anyone could help me plot the probability distribution as a function of $j$ for fixed $N$ and $t$. Please note that the probability is zero for odd $j+t$. Thank You!