the intervals $\Delta z_j$ to small (large) values. More concretely, we set $\Delta z_j=(T/2)/(4J/5)$ for $j=1, 2, \cdots, 4J/5$ and $\Delta z_j=(T/2)/(J/5)$ for $j=4J/5+1, 4J/5+2, \cdots, J$. (So we set $J$ to multiples of 5.) We use the fourth-order Runge-Kutta method for numerically solving the Schrödinger equation, where the time steps are set to about $0.1K^{-1}$. These time steps are defined by dividing $\Delta z_j$ by appropriate integers.

As mentioned in the main text, We set $J=80$ in the present simulations. The following results show that this value of $J$ is sufficiently large. Figure \[fig-J\] shows the $J$ dependence of the final photon number, $n_{\mathrm{out}}=\sum_{j=1}^J \langle \tilde{b}_j^{\dagger} \tilde{b}_j \rangle$, in the output mode. These data, the circles in Fig. \[fig-J\], are well fitted with $n_0 - b/J$, the solid lines in Fig. \[fig-J\], where $n_0$ and $b$ are the fitting parameters. This is natural because the position discretization is the first-order approximation with respect to $J^{-1}$. The dashed lines in Fig. \[fig-J\] show $n_{\mathrm{out}}=n_0$, which are the estimated values of $n_{\mathrm{out}}$ in the limit $J \to \infty$. From the small discrepancies between the data and $n_0$, indicated by arrows in Fig. \[fig-J\], the numerical errors due to finite $J$ are 