amplitude, $K$ is the magnitude of the Kerr coefficient [@comment-Kerr], $\Delta=\omega_{\mathrm{KPO}}-\omega_p/2$ is the detuning frequency ($\omega_{\mathrm{KPO}}$ is the one-photon resonance frequency of the KPO), $b^{\dagger} (\omega)$ and $b(\omega)$ are the creation and annihilation operators for photons of frequency (or wave number) $\omega_p/2+\omega$ in the output mode, and $\kappa_{\mathrm{ex}}$ is the energy decay rate of the KPO due to its coupling to the output mode. Here we assume no internal loss of the KPO, which is discussed later. Hereafter, we consider the resonance case ($\Delta=0$).

If the KPO is a closed system (${\kappa_{\mathrm{ex}}=0}$), a cat state ${|\alpha_0 \rangle + |{-\alpha_0} \rangle}$ can be generated from the vacuum state $|0\rangle$ via quantum adiabatic evolution by gradually increasing $p(t)$ from zero to ${p_0=K \alpha_0^2}$ [@Goto2016a; @Puri2017a]. When the KPO is coupled to the output mode, the photons inside the KPO will leak to the output mode. As a result, the entanglement between the KPO and the output mode arises during the generation. Moreover, the decay of the KPO due to the leak may degrade the adiabatic cat-state generation. Thus, it is not obvious whether or not we can generate a traveling cat state using the KPO.

Proposed method
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Our idea is based 