shortcut-to-adiabaticity technique proposed in Ref. 15. The technique proposed in Ref. 15 also uses an imaginary pump amplitude $p'(t)$, but it is defined as $$\begin{aligned}
p'(t)
=\frac{2\dot{\alpha}_0 \sqrt{1-2e^{-2|\alpha_0|^2}}}{1+2\alpha_0}
=
\frac{\dot{p} \sqrt{1-2e^{-2p/K}}}{\sqrt{Kp}+2p},
\label{eq-pi-Puri}\end{aligned}$$ where $\alpha_0 = \sqrt{p/K}$ is used. While the results with the technique in Ref. 15 exhibit oscillations, the results with our technique change monotonically and the final fidelity is almost perfect. These results clearly show the usefulness of our shortcut-to-adiabaticity technique for a KPO.

![Simulation results for shortcut to adiabaticity. (a) Pump amplitudes, $p(t)$ (black long-dashed line), $p'(t)$ in our method (cyan solid line), and $p'(t)$ in Ref. 15 (red short-dashed line). (b) Average photon number in KPO, $\langle a^{\dagger} a \rangle$. (c) Fidelity between the final state and the ideal even cat state with amplitude of $\sqrt{2}$. (d) Magnification of (c) around the final time. In (b)–(d), cyan solid, red short-dashed, and black long-dashed lines correspond to our method, the method proposed in Ref. 15, and the case of no $p'(t)$ (without shortcut-to-adiabaticity technique), respectively.[]{data-label="fig-shortcut"}](shortcut.eps){width="12cm"}

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D. J. Wineland, Nobel Lecture: Superposition, entanglement, and raising 