small intervals $[z_{j-1}, z_{j}]$ ($j=1, 2, \ldots, J$), where the intervals $\Delta z_j = z_{j}-z_{j-1}$ are set to small values [@supplement]. Then, $\tilde{b}(z)$ and $f_p(z)$ are discretized as follows ($z\in [z_{j-1}, z_{j}]$): $$\begin{aligned}
\tilde{b}(-z) \sqrt{\Delta z_j} 
\to \tilde{b}_j,~
f_p(T-z) \sqrt{\Delta z_j} 
\to f_j.\end{aligned}$$ Then, the commutation relation $[\tilde{b}(z),\tilde{b}^{\dagger}(z')]=\delta (z-z')$ becomes $[\tilde{b}_j, \tilde{b}^{\dagger}_l]=\delta_{j,l}$ and the normalization condition $\displaystyle \int_{0}^T \! |f_p(z)|^2 dz =1$ becomes $\displaystyle \sum_{j=1}^J |f_j|^2 =1$. By transforming the integration with respect to $z$ to the summation with respect to $j$, we obtain $$\begin{aligned}
b_p = \sum_{j=1}^{J} f_j \tilde{b}_j,~
f_j = 
\sqrt{\frac{\langle \tilde{b}_j^{\dagger} \tilde{b}_j \rangle}
{\sum_{l=1}^{J} \langle \tilde{b}_l^{\dagger} \tilde{b}_l \rangle}}.
\label{eq-bp}\end{aligned}$$

The Hamiltonian at time $t \in [z_{j-1}, z_j]$ is given by $$\begin{aligned}
H_I(t) 
&= 
H_{\mathrm{KPO}}(t)
+
i \sqrt{\kappa_{\mathrm{ex}}}
\left[
\tilde{b}_j^{\dagger} a
-
a^{\dagger} \tilde{b}_j
\right].
\label{eq-HI}\end{aligned}$$ We numerically solve the Schrödinger equation with the Hamiltonian in Eq. (\[eq-HI\]) [@supplement]. Since the Hamiltonian includes only one of $\{ \tilde{b}_j \}$, the corresponding Schrödinger equation is simple. In the present work, we investigate the cases where $\kappa_{\mathrm{ex}}=0.2K$.

As explained above, $p(t)$ should satisfy the following two conditions: $p(t)$ is increased fast enough to adiabatically generate a cat state inside the KPO before the decay spoils it; after that, $p(t)$ is 