the cases, ${I_t \approx 10 K^{-1}=2\kappa_{\mathrm{ex}}^{-1}}$, because ${\kappa_{\mathrm{ex}} I_t}$ corresponds to the photon number of the output pulse.

\[table-result\]

Improvement by shortcut to adiabaticity
=======================================

The imperfection of the generated traveling cat state may be partially due to the leak during the initial cat-state preparation. We can speed up the preparation by setting the LPF bandwidth $B$ to a larger value, e.g., $B=K$. Then, however, nonadiabatic effects degrade the cat state. The simulation results for $B=K$ are shown in Fig. \[fig-result\](b) and the second row of Table \[table-result\]. The oscillation of $\langle a^{\dagger} a \rangle$ in Fig. \[fig-result\](b) is due to the nonadiabatic effects. Consequently, the maximum fidelity between the output state and an ideal cat state decreases to 0.930 (see Table \[table-result\]).

To mitigate the nonadiabatic effects, we can use the technique called *shortcut to adiabaticity* [@Puri2017a; @Demirplak2003a; @Campo2013a]. To maintain quantum adiabatic evolution, the shortcut-to-adiabaticity technique introduces the following counterdiabatic Hamiltonian [@Demirplak2003a; @Campo2013a]: $$\begin{aligned}
H_{\mathrm{counter}} = 
i \sum_n |\dot{\phi}_n \rangle \langle \phi_n|,\end{aligned}$$ where $|\phi_n \rangle$ is the $n$-th instantaneous eigenstate of the slowly varying Hamiltonian and the dot denotes the time derivative.

In the present case, the counterdiabatic Hamiltonian is approximately given by [@supplement; @comment-shortcut] $$\begin{aligned}
H_{\mathrm{counter}} (t)
