of the FID signals are shown in red, while the imaginary in black. The full-decoupling (left panels) and selective decoupling case (right panels). The black dashed curves in the left panels are exponential fittings to the real parts of the FID signals. The green (blue) curves in the right panels are calculated real (imaginary) part of the FID signals [@Binho2019]. The blue curves overlap the experimental data and thus hardly visible. []{data-label="fig:fid"}](fid181112.pdf){width="8.6cm"}

In order to to observe the above subtle non-exponential dynamics, let us re-examine the thermal state $\rho_{\rm th}$ of the three $^{13}$C’s which is given as [@Cory1634; @Gershenfeld350; @liquidNMRQC] $$\begin{aligned}
 \label{eq_thermal_state}
\rho_{\rm th}
&\approx & 
\underbrace{\frac{ \sigma_0 + \epsilon |0\rangle \langle 0 |}{2}}_{\rm CC}
\otimes \underbrace{\frac{\sigma_0}{2}\otimes\frac{\sigma_0}{2}}_{\rm CSs} 
\nonumber \\
&+&
\underbrace{\frac{ \sigma_0}{2}}_{\rm CC}
\otimes \underbrace{
\frac{ \sigma_0 + \epsilon |0\rangle \langle 0 |}{2}
\otimes\frac{\sigma_0}{2}}_{\rm CSs} \nonumber\\
&+&
\underbrace{\frac{ \sigma_0}{2}}_{\rm CC}
\otimes \underbrace{
\frac{\sigma_0}{2} \otimes 
\frac{ \sigma_0 + \epsilon |0\rangle \langle 0 |}{2}}_{\rm CSs} .\end{aligned}$$ Here, $\epsilon %\sim \hbar \gamma_{\rm G} B/k_B T 
\sim 10^{-5}$ in NMR measurements. When we observe only CC, $\rho_{\rm th}$ is equivalent to $$\begin{aligned}
 \rho_{\rm th}
\approx
\frac{1}{8}
\left( \sigma_0 + \epsilon |0\rangle \langle 0 | \right)  
\otimes \left( |0\rangle\langle 0|+ |1\rangle\langle 1|\right)
\otimes 