D(-\beta) \rho D(\beta) P \right]}$ [@Leonhardt; @Deleglise2008a; @Goto2016a], where $\displaystyle {D(\beta)=\exp \! \left( \beta b_p^{\dagger} - \beta^* b_p \right)}$ (the asterisk denotes complex conjugation) and $\displaystyle {P=\exp \! \left( i\pi b_p^{\dagger} b_p \right)}$.

The results of the numerical simulation are shown in Fig. \[fig-result\](a) and the first row of Table \[table-result\]. Table \[table-result\] also provides the setting of the simulation. Figure \[fig-result\](a) shows that the photon number in the KPO varies in a similar manner to the pump amplitude, as expected. The Wigner function for the output pulse in Fig. \[fig-result\](a) clearly shows the interference fringe, which is the evidence for the quantum superposition of the two coherent states, that is, the output pulse is in a cat state. (The tilt of the Wigner function, which corresponds to $\theta_{\mathrm{cat}}$ in Table \[table-result\], is due to the residual Kerr effect.) As shown in Table \[table-result\], the maximum fidelity between this output state and an ideal cat state is 0.962. Thus, the present method works successfully as expected.

![Simulation results of traveling cat-state generation using a KPO. Left: Time evolutions of the pump amplitude $p(t)$ and the expectation value of the photon number, $\langle a^{\dagger} a \rangle$, in the KPO. Right: Wigner 