\left( |0\rangle\langle 0|+ |1\rangle\langle 1|\right).\end{aligned}$$ Moreover, $\sigma_0$ of CC is not observable in NMR and thus $\rho_{\rm th}$ can be re-normalized as $$\begin{aligned}
 \rho_{\rm th}
\approx
\frac{1}{8}
\left( |0\rangle \langle 0 | \right)  
\otimes \left( |00\rangle\langle 00| + |01\rangle\langle 01|
+|10\rangle \langle 10| + |11\rangle \langle 11| \right).\end{aligned}$$ The interaction effects on CC from the $ |01\rangle\langle 01|$ and $|10\rangle \langle 10|$ states of CSs cancel with each other and thus if we can prepare $$\begin{aligned}
\label{eq_rho_i}
 \rho_{\rm i} &=&  |+\rangle \langle + | \otimes 
\left( 
|01\rangle\langle 01|+|10\rangle \langle 10| 
\right),\end{aligned}$$ we are able to observe the subtle non-exponential dynamics discussed before. Here, $\displaystyle |+\rangle = 
\frac{|0\rangle + |1\rangle}{\sqrt{2}}$. This $\rho_{\rm i}$ can be prepared with the standard NMR technique called a soft pulse [@Levitt2008].

The results are summarized in Fig. \[fig:fid\]. We can successfully observe the exponential decays in the full decoupling cases (left panels), while the non-exponential decay dynamics are observed in the selective decoupling cases (right panels). We also calculated the decay dynamics from the data summarized in Table \[FID\_tc\] as in our preceding work [@Binho2019] which are plotted as green (blue) solid curves on the right panels in Fig. \[fig:fid\]. By taking into account that there 