an output mode. The KPO is implemented by a Josephson parametric oscillator (JPO) [@Lin2014a] with a relatively large Kerr effect. The JPO is capacitively coupled to a transmission line for the output mode. Traveling cat state is generated by controlling the pump field dynamically.[]{data-label="fig-system"}](system.eps){width="8cm"}

The coupled system of a KPO and an output mode is depicted in Fig. \[fig-system\], where a superconducting-circuit implementation is supposed [@Nigg2017a; @Puri2017b; @Lin2014a]. In a frame rotating at half the pump frequency, $\omega_p /2$, of the parametric pumping and in the rotating-wave approximation, the system is modeled by the following Hamiltonian [@Goto2016a; @Puri2017a; @Gardiner1985a; @Milburn; @Duan2003a; @Goto2005a] (we use the units $\hbar = v_p =1$, where $v_p$ is the phase velocity of the electromagnetic fields in the output mode): $$\begin{aligned}
H(t)
&=
H_{\mathrm{KPO}}(t)+H_{\mathrm{out}}+H_{c},
\label{eq-H}
\\
H_{\mathrm{KPO}}(t)
&=
\frac{p(t)}{2} 
\left(
a^{\dagger 2} + a^2
\right)
-
\frac{K}{2} a^{\dagger 2} a^2
+
\Delta a^{\dagger} a,
\label{eq-HKPO}
\\
H_{\mathrm{out}}
&=
\int_{-\infty}^{\infty} \! \omega b^{\dagger} (\omega ) b(\omega ) \, d\omega,
\label{eq-Hout}
\\
H_c
&=
i 
\sqrt{\frac{\kappa_{\mathrm{ex}}}{2\pi}}
\int_{-\infty}^{\infty} \! 
\left[
b^{\dagger} (\omega ) a - a^{\dagger} b(\omega )
\right] d\omega,
\label{eq-Hc}\end{aligned}$$ where $a^{\dagger}$ and $a$ are the creation and annihilation operators for the KPO, $p(t)$ is the time-dependent pump 