&= 
i \frac{p'(t)}{2} \left( a^{\dagger 2} - a^2 \right),
\label{eq-counter}
\\
p' (t)
&= 
\frac{\dot{p}(t)}{p(t)} \tanh \! \frac{p(t)}{K}.
\label{eq-pi}\end{aligned}$$ The physical meaning of the counterdiabatic Hamiltonian is to add the imaginary pump amplitude $p'(t)$ to the real one $p(t)$. This is experimentally possible by controlling the phase of the pump field.

The simulation results with the shortcut-to-adiabaticity technique are shown in Figs. \[fig-result\](c) and \[fig-result\](d) and the third and fourth rows of Table \[table-result\]. As shown in Fig. \[fig-result\](d), the oscillation of $\langle a^{\dagger} a \rangle$ does not occur even when $B=K$, unlike Fig. \[fig-result\](b). This demonstrates that the shortcut-to-adiabaticity technique works successfully. The fidelity is improved for both $B=0.5K$ and $K$. Contrary to the results without the shortcut-to-adiabaticity technique, the fidelity is higher for larger $B$, and the corresponding infidelity is lower than 1% when $B=K$ (See Table \[table-result\]). Thus, the shortcut-to-adiabaticity technique can significantly improve the quality of the traveling cat state.

Internal loss
=============

So far, internal loss of the KPO has not been taken into account. However, any actual devices have internal loss, and it degrades the coherence of the cat states. Here we briefly examine the effect of the internal loss in the superconducting-circuit implementation 