decreased as $\displaystyle {p(t) \propto \exp (-\kappa_{\mathrm{ex}} t)}$ so that the Kerr term and the pumping term are cancelled out. To satisfy these conditions simultaneously, we define $p(t)$ as the output of the fourth order low-pass filter (LPF) [@supplement; @Wiseman] with the input ${p_{\mathrm{in}}(t)=K A_p \exp (-\kappa_{\mathrm{ex}} t)}$. The dimensionless parameter $A_p$ is used for tuning the photon number of the traveling cat state. We set the photon number to about 2, which is large enough for the two coherent states being distinct but small enough to solve the Schrödinger equation numerically. The bandwidth, $B$, of the LPF [@supplement; @Wiseman] is set to $B=0.5K$ between $\kappa_{\mathrm{ex}}$ and $K$. The final time $T$ is set such that the final photon number, $n_{\mathrm{in}}$, in the KPO is less than $10^{-3}$ (see Table \[table-result\]).

We obtain the density operator, $\rho$, describing the quantum state of the output pulse using the moments $M_{m,n}=\langle b_p^{\dagger m} b_p^n \rangle$ with $b_p$ in Eq. (\[eq-bp\]). The density matrix with respect to the Fock states is given by the following formula [@Eichler2012a]: $$\begin{aligned}
\rho_{m,n}
= \frac{1}{\sqrt{m!n!}}
\sum_{l=0}^{\infty} \frac{(-1)^l}{l!} M_{n+l,m+l}.
\label{eq-rho}\end{aligned}$$ Using the density matrix, we calculate the corresponding Wigner function $\displaystyle {W(\beta ) = 
\frac{2}{\pi} \mbox{Tr} \! \left[ 