improve the quality of the traveling cat state significantly. The traveling cat states generated by a KPO can be directly observed by, e.g., homodyne or heterodyne detection of the output field. Thus, this can be used for experimentally demonstrating the ability of a KPO to generate cat states deterministically.

Acknowledgments {#acknowledgments .unnumbered}
===============

HG thanks Kazuki Koshino for his suggestion. This work was supported by JST ERATO (Grant No. JPMJER1601).

Numerical simulation method
===========================

In the simulation presented in the main text, we truncate the photon number in the output mode at 6, which is sufficiently large to express a cat state with average photon number of 2. Then, the state vector $|\psi \rangle$ describing the coupled system is represented as follows: $$\begin{aligned}
|\psi \rangle
&=
\sum_{n=0}^{N_0} \psi_0 (n) |n\rangle |0\rangle
+
\sum_{n=0}^{N_1} \sum_{j_1=1}^{J} \psi_1 (n,j_1) |n\rangle |j_1\rangle
+
\sum_{n=0}^{N_2} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \psi_2 (n,j_1,j_2) |n\rangle |j_1,j_2\rangle
\nonumber
\\
&+
\sum_{n=0}^{N_3} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} 
\psi_3 (n,j_1,j_2,j_3) |n\rangle |j_1,j_2,j_3\rangle
+
\sum_{n=0}^{N_4} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} \sum_{j_4=1}^{j_3} 
\psi_4 (n,j_1,j_2,j_3,j_4) |n\rangle |j_1,j_2,j_3,j_4\rangle
\nonumber
\\
&+
\sum_{n=0}^{N_5} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} \sum_{j_4=1}^{j_3} \sum_{j_5=1}^{j_4} 
\psi_5 (n,j_1,j_2,j_3,j_4,j_5) |n\rangle |j_1,j_2,j_3,j_4,j_5\rangle
\nonumber
\\
&+
\sum_{n=0}^{N_6} \sum_{j_1=1}^{J} \sum_{j_2=1}^{j_1} \sum_{j_3=1}^{j_2} \sum_{j_4=1}^{j_3} \sum_{j_5=1}^{j_4} \sum_{j_6=1}^{j_5} 
\psi_6 (n,j_1,j_2,j_3,j_4,j_5,j_6) |n\rangle |j_1,j_2,j_3,j_4,j_5,j_6\rangle,
\label{eq-psi}\end{aligned}$$ where the first and second 