estimated to be less than 1%. This indicates that $J$ is sufficiently large.

![$J$ 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. (a)–(d) correspond to Figs. 2(a)–2(d), respectively, in the main text. Circles represent simulation results. Solid lines are fitted curves with $n_{\mathrm{out}}(J) = n_0 - b/J$, where $n_0$ and $b$ are the fitting parameters. Horizontal dashed lines represent $n_{\mathrm{out}}(J) = n_0$. Arrows indicate the discrepancies between the data and $n_0$.[]{data-label="fig-J"}](J.eps){width="12cm"}

Pulse shape control using low-pass filters
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In the simulation presented in the main text, we define the pulse shape of the pump amplitude $p(t)$ as the output of the fourth-order low-pass filter (LPF) \[29\] with the input ${p_{\mathrm{in}}(t)=K A_p \exp (-\kappa_{\scriptsize{\mbox{ex}}} t)}$. The input-output relation of a LPF is given by $p_{\mathrm{out}}(t)
=\int_0^t \! B e^{-B(t-s)} p_{\mathrm{in}}(s) \, ds$ \[29\], where $B$ is the bandwidth of the LPF and $p_{\mathrm{in}}(s)=0$ (${s<0}$) is assumed. Note that $\dot{p}_{\mathrm{out}}(t)
=-B [p_{\mathrm{out}}(t) - p_{\mathrm{in}}(t)]$, where the dot denotes the time derivative. Thus, we can calculate the output of the LPF by numerically solving this differential equation. The $n$-th order LPF is defined as the output of the LPF the input of which is the output 