function, $W(\beta )$, of the output pulse. See Table \[table-result\] for the settings of (a)–(d).[]{data-label="fig-result"}](result.eps){width="15cm"}

                             Fidelity    $~~\beta_{\mathrm{cat}}^2~~$   $~~\theta_{\mathrm{cat}}/\pi~~$    $n_{\mathrm{in}}$    $~~K I_t~~$    Shortcut    $~~\kappa_{\mathrm{ex}}/K~~$   $~~B/K~~$   $~~A_p~~$   $~~KT~~$   $~~J~~$
  ------------------------ ------------ ------------------------------ --------------------------------- --------------------- ------------- ------------ ------------------------------ ----------- ----------- ---------- ---------
  Fig. \[fig-result\](a)      0.962                  2.01                            0.03                 $6.2\times 10^{-4}$      9.63         Unused                 0.2                   0.5        2.45         50        80
  Fig. \[fig-result\](b)      0.930                  1.96                            0.02                 $6.1\times 10^{-4}$      9.64         Unused                 0.2                   1.0        2.15         45        80
  Fig. \[fig-result\](c)      0.983                  2.03                            0.02                 $6.4\times 10^{-4}$      9.63          Used                  0.2                   0.5        2.50         50        80
  Fig. \[fig-result\](d)      0.993                  2.02                            0.01                 $6.4\times 10^{-4}$      9.60          Used                  0.2                   1.0        2.25         45        80

  : Results and setting of numerical simulations. Fidelity: maximum fidelity between the output state and the ideal cat state with two parameters: $\displaystyle (|\beta_{\mathrm{cat}} e^{i\theta_{\tiny{\mbox{cat}}}} \rangle
      +|{-\beta_{\mathrm{cat}} e^{i\theta_{\tiny{\mbox{cat}}}}} \rangle )
      /\sqrt{2(1+e^{-2\beta_{\tiny{\mbox{cat}}}^2})}$, where $\beta_{\mathrm{cat}}$ and $\theta_{\mathrm{cat}}$ are the magnitude and phase of the cat-state amplitude, respectively. $\beta_{\mathrm{cat}}^2$ and $\theta_{\mathrm{cat}}/\pi$: values maximizing the fidelity. $\beta_{\mathrm{cat}}^2$ is close to 2, which means that the photon number of the output pulse is about 2. $n_{\mathrm{in}}$: average photon number in the KPO at the final time $T$. $I_t$: time integral of the average photon number in the KPO, that is, $I_t=\int_0^T \! \langle a^{\dagger} a \rangle \, dt$. In all 